Sum Of Sequence Calculator

Use this calculator to determine the future value of your savings and lump sum. In geometric progressions where |r| < 1 (in other words where r is less than 1 and greater than –1), the sum of the sequence as n tends to infinity approaches a value. This calculator will find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). The Arithmetic series of finite number is the addition of numbers and the sequence that is generally followed include – (a, a + d, a + 2d, …. Approach : Read input number asking for length of the list using input() or raw_input(). Sum of 32-bit integer quantities is not computed by using 64-bit results, and overflow can occur for the LINQ to SQL translation of Sum. This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. Sequences and summations CS 441 Discrete mathematics for CS M. Objects might be numbers or letters. Set Sum = 0. The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Partial Sums of an Arithmetic Sequence. All term in the sequence meet a specific logical rule which needs to be recognised in order to find the missing terms. Because a series consists of evenly-spaced numbers, the median and mean (average) of the series will be the same. arithmetic series. This way, each term can be expressed by this equation: xₐ = xₐ₋₁ + xₐ₋₂. ): A series of numbers in which ratio of any two consecutive numbers is always a same number that is constant. Calculates the sum of the infinite geometric series. Arithmetic Sequences and Sums Sequence. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. We start with the formula for PV of a future value (FV) single lump sum at time n and interest rate i,. lets say you want to sum the numbers from 1 to 100 all you have to do is to calculate this (100)x(101) /2. What Is The Formula For Calculating Arithmetic Sequence? If the initial term of an arithmetic sequence is a 1 and the common difference of successive members is d , then the nth term of the sequence is given by:. The sequence begins 1, 1, 2, 3, 5, and each succeeding term is the sum of the previous two terms. If only a single number for value1 is supplied, SUM returns value1. Example 1: Find the sum of squares of the numbers from 0 to 5000. I want to calculate sum of the average daily value off each product. Work out: a. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. skipna bool, default True. The Corbettmaths video tutorial on finding the nth term for a fractional sequence. In this case, you have the sequence. Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums. P Class 11 Engineering - Sum of n Terms of G. The formula for the n-th term of a quadratic sequence is explained here. And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs. The sequence of terms that are common to both sequences will be an arithmetic sequence with a common difference of 28, which is the LCM of 4 and 7. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. Add Number to Sum and set equal to Sum. This calculator will find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). What is Sum of the Series in Math. The arithmetic sequence calculator uses arithmetic sequence formula to find sequence of any property. The problem is that the number of cells to be summed will vary; for one run of the macro it could be 100 cells, while on the next it could be 300 and on the third only 25. std ([ddof]) Calculate window standard deviation. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. Geometric Progression (G. Sequence: By a sequence, we mean an arrangement of number in definite order according to some rules. Build your own widget. Improve your math knowledge with free questions in "Find the sum of an arithmetic series" and thousands of other math skills. The sum of a geometric series is indeed an interesting place to start this discussion. You also generate a sum in each step, which you reuse in the next step. Home; Java Tutorial; Language; Data Type; Operators; Statement Control. Find the sum of first n terms of the series (i) 3 +33 +333 +. Then we update the value of variable sum as - sum = sum + c = 0 + 0 = 0. To concatenate a series of iterables, consider using itertools. help me how to calculate the sum of a series in Matlab. The Fibonacci sequence typically has first two terms equal to x₀ = 0 and x₁ = 1. In other words, how to take the value of a cell located in one worksheet and add it to the value of another cell located in another worksheet to come up with the total of the respective cells. The sequence begins 1, 1, 2, 3, 5, and each succeeding term is the sum of the previous two terms. Calculate the sum of series 1^2+2^2+3^2+4^2++n^2 (n>0) using the both for and while loop structure. Subtract the smaller fraction from the larger fraction to calculate the. Sum of 32-bit integer quantities is not computed by using 64-bit results, and overflow can occur for the LINQ to SQL translation of Sum. The partial sum approach of course involves a "trick" as well -- finding an expression for the dang partial sum. To find sum of your series, you need to choose the series variable, lower and upper bounds and also input the expression for n-th term of the series. Example 2: Find the sum of squares of the first 100 numbers of the form prime minus one. This calculator for to calculating the sum of a series is taken from Wolfram Alpha LLC. As n tends to infinity, S n tends to The sum to infinity for an arithmetic series is undefined. Leonhard Euler was able to calculate the exact sum of the p-series with p 2: 2-2 32 42 Use this fact to find the sum of each series: 2 32 so. Recently, mandatory vote-by-mail has received a great deal of attention as a means of administering elections in the United States. 01 then we use the formula for the sum of the infinite geometric series S oo = a 1 / (1 - r),. Using for loop. Program description:- Find sum of series 1² + 2² + 3² + 4² + 5² +…. com presented a large number of task in mathematics that you can solve online free of charge on a variety of topics: calculation of integrals and derivatives, finding the sum of the series, the solution of differential equations, etc. You may find functions such as sum, ones, or dot useful. In other words, how to take the value of a cell located in one worksheet and add it to the value of another cell located in another worksheet to come up with the total of the respective cells. sum (axis = None, skipna = None, level = None, numeric_only = None, min_count = 0, ** kwargs) [source] ¶ Return the sum of the values for the requested axis. If the value of the sum (in the limiting sense) exists, then they say that the series. Calculate the first term. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson arithmetic sequence. Our sum of series calculator or arithmetic series calculator is an online tool which you can find on Google. is 1 and any term is equal to the sum of all the succeeding terms. It is used to achieve the gas–liquid or liquid–liquid two-phase separation. NPV = Sum CF* ((1+2%)/(1+D))^N When N is infinite, after simplification, NPV = CF * 1 / (1 - r) where r = (1+2%)/(1+D)--A+ V. In general, one does not expect to be able to calculate an infinite sum exactly. In this case, you have the sequence. But there are some series. By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. Java program to calculate the sum of N numbers using arrays, recursion, static method, using while loop. Each number in series is called as Fibonacci number. Leonhard Euler continued this study and in the process solved many. Finding a Rule for a Sequence [07/24/2003] What is the next number in this sequence? 1, 3, 11, 67, ? Finding Sum Formula using Sequences of Differences [06/28/1998] Finding a formula for the sum of the first n fourth powers using sequences of differences. 02 Line 10 Column 10 "The Sum of the above Series for term "s is : ". Find the sum of 35 terms of an arithmetic series of which the first term is "a" and the fifteenth term is "9a. It is said that when Gauss was ten. We start with the formula for PV of a future value (FV) single lump sum at time n and interest rate i,. I evaluated the partial sum through a calculator, my answer was -19/30, or -. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. After discussing the above two examples one will wonder if any sequence has the same faith (meaning, it gets closer to a number). Sum of this series is calculated by the formula : sum=(n*(n+1))/2. It will also check whether the series converges. So, the given series is an arithmetic progression whose common difference is d = 2. An arithmetic sequence is one in which the difference between successive members is a constant. demonstration: form input vectors for 10, ten thousand, and 1*10^(many) 0 10 3 0 10 5 0 10 15 0 10000. We therefore derive the general formula for evaluating a finite arithmetic series. The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Geometric Progression, Series & Sums Introduction. By applying this calculator for Arithmetic & Geometric Sequences, the n-th term and the sum of the first n terms in a sequence can be accurately obtained. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. The T-junction is a novel type of separator used in the petroleum and gas industry. What two things do you need to know to find the sum of an infinite geometric series? Find the sum of the infinite geometric series. asked by ryan on March 15, 2017; Algebra. Explain why or why not. Please help. The two sequences are placed in two consecu- tive rows. The sums we have looked at so far are finite sums with finite upper and lower limits. I know the inequality of [integral from n+1 to infinity of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Thanks for your help. You need to know both of these numbers in order to calculate the sum of the arithmetic sequence. 1 - Enter the first term A1 in the sequence, the common ratio r and n n the number of terms in the sum then press enter. Hope this helps!. S = 10 1 2, a 1= 1 2 Write. +N^2 in C Programming Language. Such series appear in many areas of modern mathematics. This formula shows that a constant factor in the summands can be taken out of the sum. Free Summation Calculator. However, when the series has an infinite number of terms the summation is more complicated and the series may or may not have a finite. Let the variable equal the first term in the sequence, and equal the last term in the sequence. Observe that for the geometric series to converge, we need that $$|r| 1$$. Excel's Calculation Process. Any sequence that has a common second difference is a quadratic sequence. So the second term common to both sequences is 1+28 = 29. arithmetic series. asked by ryan on March 15, 2017; Algebra. Infinite Series calculator is a free online tool that gives the summation value of the given function for the given limits. In the opposite case, one should pay the attention to the «Series convergence test» pod. Sum of series in C language 1² + 2² + 3² + 4² + 5² +…. Linear sequences A number pattern which increases (or decreases) by the same amount each time is called a linear sequence. Sequences and summations CS 441 Discrete mathematics for CS M. s n = 2 − 3(0. Get comfortable with sequences in general, and learn what arithmetic sequences are. com Task : To find the sum of all the elements in a list. Calculates the 8-bit checksum for a sequence of hexadecimal bytes. On our site OnSolver. I understand the process in calculating a simple series like this one $$\\sum_{n=0}^4 2n$$ but I do not understand the steps to calculate a series like this one $$\\sum_{n=1}^x n^2$$ I have an aw. Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums. Learn Java by Examples: Write a program to calculate and print the sum of the following series: Sum(x) = 2 – 4 + 6 – 8 + 10 - 12 … - 100Learn Java by examples. See full list on mathsisfun. The free tool below will allow you to calculate the summation of an expression. S = 12, a 1= 2 8. This Arithmetic Sequence Calculator is used to calculate the nth term and the sum of the first n terms of an arithmetic sequence. First simplify the expression, X! receives great quick! x/x! = x/(x * (x-a million) * (x-2) *a million) hence x/x! = a million/(x-a million)! for x >= 2 and a million for x =a million Then note that that's the same to the enlargement of the Taylor series for e^x as a million + x + x^2/2! + x^3/3! +x^4/4! for x = a million. The sequence of partial sums of a series sometimes tends to a real limit. If you're behind a web filter, please make sure that the domains *. Free series convergence calculator - test infinite series for convergence step-by-step This website uses cookies to ensure you get the best experience. This is a free javascript calculator. Leonhard Euler continued this study and in the process solved many. \$1000000007\$ is a prime number, so division by \$2^L\$ modulo 1000000007 is a multiplication by a multiplicative inverse of 2 modulo 1000000007 (which is trivial to find) to the same power. 99805 Time Complexity: O(n). In general, one does not expect to be able to calculate an infinite sum exactly. As the top row increases, the bottom row decreases, so the sum stays the same. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. The T-junction is a novel type of separator used in the petroleum and gas industry. Sum definition is - an indefinite or specified amount of money. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. The first sum in the numerator is the sum of squared residuals for the first model (e. It is used to achieve the gas–liquid or liquid–liquid two-phase separation. On a higher level, if we assess a succession of numbers, x 1 , x 2 , x 3 ,. This document explains how to calculate the sum or total when working with cell data located in multiple worksheets. Explain why or why not. Finding a general expression for a partial sum by induction and then finding the limit of this partial sum is a perfectly valid technique. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Improve your math knowledge with free questions in "Find the sum of an arithmetic series" and thousands of other math skills. Get comfortable with sequences in general, and learn what arithmetic sequences are. var ([ddof]) Calculate unbiased window variance. 2, ALPHA ([I], 1, 1 0) EXE, and you should get an answer of 9. The z-transform of the unit pulse, = 1. Learn Java by Examples: Write a program to calculate and print the sum of the following series: Sum(x) = x/2 + x/5 + x/8 + … + x/100. BYJU'S online infinite geometric series calculator tool makes the calculation faster, and it displays the sum in a fraction of seconds. Sequences and summations CS 441 Discrete mathematics for CS M. Geometric Progression (G. Sum of Natural, Odd or Even Numbers Series Calculator getcalc. Java program to calculate the sum of GP series. Just enter the expression to the right of the summation symbol (capital sigma, Σ) and then the appropriate ranges above and below the symbol, like the example provided. This summation notation calculator can sum up many types of sequencies including the well known arithmetic and geometric sequencies, so it can help you to find the terms including the nth term as well as the sum of the first n terms of virtualy any series. Series (Find the sum) A finite Geometric Series (a limited number of terms, or Partial Sum) An infinite Geometric Series, if our infinite series is. Solving mathematical problems online for free. help me how to calculate the sum of a series in Matlab. The sums we have looked at so far are finite sums with finite upper and lower limits. To evaluate the Midpoint Sum, it gets messier. For some use cases, there are good alternatives to sum(). The sum of the members of a finite arithmetic progression is called an arithmetic series. Sn = (n/2) [a + l] Sn = (n/2) [2a + (n - 1)d] a = first term, n = number of terms of the series, d = common difference and l = last term. 0 / k for k in range(1, 10001)) What this code does: the innermost part is a generator expression, which computes the elements of a series 'on the fly'. Example 1: Find the sum of squares of the numbers from 0 to 5000. For example, in the sequence 10, 15, 20, 25, 30. Observe that for the geometric series to converge, we need that $$|r| 1$$. So again, a problem about earned interest might not be a perfect example, since you can withdraw your money at any instant and not only at whole number year values. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. 1 26) a 12 = 28. Arithmetic Sequence. \) The series becomes $$\sum\limits_{n = 0}^\infty {\large\frac{{{u^n}}}{{n!}} ormalsize}. More Practice Problems with Arithmetic Sequence Formula Direction: Read each arithmetic sequence question carefully, then answer with supporting details. Given real (or complex!) numbers aand r, X1 n=0 arn= (a 1 r if jr <1 divergent otherwise The mnemonic for the sum of a geometric series is that it’s \the rst term divided by one minus the common ratio. Another approach could be to use a trigonometric identity. The sums we have looked at so far are finite sums with finite upper and lower limits. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson arithmetic sequence. the starting principal you'll need to achieve the payouts desired:. org are unblocked. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. In column B, create the sequence that you are going to sum and name it seqa. A sequence is a list of numbers. After discussing the above two examples one will wonder if any sequence has the same faith (meaning, it gets closer to a number). Free Summation Calculator. [Note that the sequence of the sums S 1 (the sum of the first term), S 2 (the sum of the first two terms), S 3 (the sum of the first three terms),. The present value, PV, of a series of cash flows is the present value, at time 0, of the sum of the present values of all cash flows, CF. To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. Only this variable may occur in the sequence term. is 16 and sum of the next 3 terms is 128. The denominator doesn't change. This calculator for to calculating the sum of a series is taken from Wolfram Alpha LLC. How to use sum in a sentence. Because 1 is paired with 10 (our n), we can say that each column has (n+1). Alex's Arithmetic Sequence Sum Calculator is a very simple program, which allows you to get the sum of an Arithmetic Squence, it supports two types of sequences: ￭ type 1 - if you have the first. When the denominators are the same, add the numerators, or top numbers, to calculate the answer. arithmetic series. In column A, create the sequence 1, 2, 3,. ID 1001746465 1 1 1 1 23 4+5 1 There are several ways to implement this. Sum of Three Consecutive Integers calculator. I hope it is enough to get you going in the right direction. The sum of geometric series refers to the total of a given geometric sequence up to a specific point and you can calculate this using the geometric sequence solver or the geometric series calculator. Series calculator allows to calculate online the sum of the terms of the sequence whose index is between the lower and the upper bound. My dxp and csv file is uploaded in below. If you are struggling to understand what a geometric sequences is, don't fret! We will explain what this means in more simple terms later on and take a look at. Limit Comparison Test If lim (n-->) (a n / b n) = L, where a n, b n > 0 and L is finite and positive,. This value is equal to:. Solution: Given decimal we can write as the sum of 0. Leonhard Euler was able to calculate the exact sum of the p-series with p 2: 2-2 32 42 Use this fact to find the sum of each series: 2 32 so. Notice \t is used to provide 8 spaces (1 tab) between two values see in the output. It should be noted, that if the calculator finds sum of the series and this value is the finity number, than this series converged. hi i just want to confirm my method of using sum of pairs is correct. Because the Fibonacci value for 20000 has 4179 decimals and it needs quite an impressive amount of processing, the maximum allowed value is 20000. 0 / k for k in range(1, 10001)) What this code does: the innermost part is a generator expression, which computes the elements of a series 'on the fly'. As running variable, which is increased by 1 in each step, i is used. Excel does not calculate cells in a fixed order, or by Row or Column. We used a sum-over-states method to calculate the dynamic polarizabilities of 6S 1/2 ground state and highly-excited (nS 1/2 and nP 3/2 ) Rydberg state of cesium atoms, and identify corresponding magic detuning for optical wavelengths in the range of 850-1950 nm. Z Transforms of Common Sequences Ele 541 Electronic Testing Unit Pulse. We call an a term of the sequence. If you're seeing this message, it means we're having trouble loading external resources on our website. Representations of N as a sum of distinct elements from special sequences D. What Is Arithmetic Sequence? is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. If the series has a finite number of terms, it is a simple matter to find the sum of the series by adding the terms. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. this, use your calculator and examine high powers of numbers between 1 and -1. A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The Corbettmaths video tutorial on finding the nth term for a fractional sequence. A series, which is not a list of terms like a sequence, is the sum of the terms in a sequence. A finite number of terms of an arithmetic sequence can be added to find their sum. Often the first numbers will be 1, but not always. If S n tends to a limit as n tends to infinity, the limit is called the sum to infinity of the series. But there are some series. I evaluated the partial sum through a calculator, my answer was -19/30, or -. Exclude NA/null. In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. How to Find Sum of First n Terms When nth Term is Given ? A series whose terms are in Arithmetic progression is called Arithmetic series. Sum of this series is calculated by the formula : sum=(n*(n+1))/2. The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. ID 1001746465 1 1 1 1 23 4+5 1 There are several ways to implement this. Work out: a. day, sum((random()*5)::integer) num FROM days -- left join other tables here to get counts, I'm using random group by days. The Sum (Summation) Calculator is used to calculate the total summation of any set of numbers. P Class 11 Engineering - Sum of n Terms of G. But there are some series. Calculate sum elements of sequence: sum. Doing so means that most of the depreciation associated with an asset is recognized in the first few years of its useful life. All term in the sequence meet a specific logical rule which needs to be recognised in order to find the missing terms. Thus the message becomes: Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1. The first term of an infinite G. After all, yes 1/(1-x) has an honest-to-goodness explosion to infinity at x=1, but it makes perfectly good sense at x=-1, and tells us (what my calc students. Sum of elements in a list — programminginpython. Series (Find the sum) When you know the first and last term. [ Don't peek. Recently, mandatory vote-by-mail has received a great deal of attention as a means of administering elections in the United States. Write a program in C++ to calculate the sum of the series (1*1) + (2*2) + (3*3) + (4*4) + (5*5) + + (n*n). 02 Line 10 Column 42 Pic zz9 From N Highlight. I know the inequality of [integral from n+1 to infinity of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Number sequences questions usually consist of four to seven visible numbers along with a single missing number or, depending on the sequence's complexity level, 2 or 3 missing numbers. An infinite series has an infinite number of terms. Number sequences test practice for aptitude tests and psychometric IQ tests. The sum of a sequence is known as a series, and the harmonic series is an example of an infinite series that does not converge to any limit. ${s_n} = \sum\limits_{i = 1}^n i$. Sum of: from: to: Submit: Share a link to this widget: More. Solution: A series in which each number is sum of its previous two numbers is known as Fibonacci series. mean (*args, **kwargs) Calculate the window mean of the values. 23) a 21 = −1. The program solves Riemann sums using one of four methods and displays a graph when prompted. Fourth, we recall the sum of the X 2 and subtract 240. ID 1001746465 1 1 1 1 23 4+5 1 There are several ways to implement this. To be a good Java developer is to be fluent in Java 8. Sum of this series is calculated by the formula : sum=(n*(n+1))/2. Just enter the expression to the right of the summation symbol (capital sigma, Σ) and then the appropriate ranges above and below the symbol, like the example provided. The nineteenth term. 2, ALPHA ([I], 1, 1 0) EXE, and you should get an answer of 9. An arithmetic sequence is one in which the difference between successive members is a constant. To use this calculator, simply type in your list of inputs separated by commas (ie 2,5,8,10,12,18). Print the sum. Sum of sequence calculator. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. Geometric Progression (G. asked by ryan on March 15, 2017; Algebra. For the finite sums series calculator computes the answer quite literally, so if there is a necessity to obtain a short expression we recommend computing a parameterized sum. This website uses cookies to ensure you get the best experience. Infinite Series calculator is a free online tool that gives the summation value of the given function for the given limits. I have made this small program to help me add numbers and find the sum, it runs but no output. The nth term of a sequence is given by U n = n 2 /(n + 1). 02 Line 10 Column 58 Pic zz. Everything you want to know about Java. I know the inequality of [integral from n+1 to infinity of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson arithmetic sequence. P Class 11 Engineering - Sum of n Terms of G. Write a program in C++ to calculate the sum of the series (1*1) + (2*2) + (3*3) + (4*4) + (5*5) + + (n*n). The sequence of these partial sums converges to also. 78 degrees Fahrenheit. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. com allows you to find the sum of a series online. For help creating a sequence, see Note 1I. If the range of a sum is finite, is typically assigned a sequence of values, with being evaluated for each one. Third, we square the sum of X (45 times itself = 2025) and divide it by N (number of scores). A finite number of terms of an arithmetic sequence can be added to find their sum. The 8-bit checksum is the 2's complement of the sum off all bytes. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. 01 then we use the formula for the sum of the infinite geometric series S oo = a 1 / (1 - r),. Such series appear in many areas of modern mathematics. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. #sum_(k=1)^nk=(n(n+1))/2# and. The amount it increases or decreases by is known as the common difference. Add the numbers and divide by (n - 1) = 6 to get 95. "Mike" wrote: > I have a series of cashflows, forecast to grow at say 2% each year and > go on indefinitely. Objects might be numbers or letters. Using the nth term. Sum of a Series: If we are able to find the general term of the series successfully, then we can also. Work out: a. Which of the pairs of events below is dependent? Select the correct answer below: drawing a 7 and then drawing another 7 with replacement from a standard deck of cards rolling a 1 and then rolling a 6 with a standard die rolling a 3 and then rolling a 4 with. The sequence of these partial sums converges to also. Thus the message becomes: Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1. An example of the sequence can be seen as follows:. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. BYJU’S online infinite series calculator tool makes the calculations faster and easier where it displays the value in a fraction of seconds. It will also check whether the series converges. The numerator above is a difference of two partial sums of the \2^n*a_{n}\ sequence. The two sequences are placed in two consecu- tive rows. This technique causes problems in several situations, however and cannot be universally relied upon. Using Wolfram Alpha 214 , put in 10 x x and you will get can be reduced to the much easier-to-calculate infinite sum 1 - x - x2 Series - A series is formed by the sum or addition of the terms in a two equations that allow easy calculation for an arithmetic series. The numerator above is a difference of two partial sums of the \2^n*a_{n}\ sequence. More Practice Problems with Arithmetic Sequence Formula Direction: Read each arithmetic sequence question carefully, then answer with supporting details. day, sum((random()*5)::integer) num FROM days -- left join other tables here to get counts, I'm using random group by days. For example, two-thirds plus four-thirds equals six-thirds. Average = Sum of terms / Number of terms. is 16 and sum of the next 3 terms is 128. Methods for Evaluating In nite Series Charles Martin March 23, 2010 Geometric Series The simplest in nite series is the geometric series. The sum of the first three terms of the sequence of terms common to the two given sequences is 3 times the second term of the sequence. The sum of geometric series refers to the total of a given geometric sequence up to a specific point and you can calculate this using the geometric sequence solver or the geometric series calculator. interval is [FIRST, LAST) NB. The sum of the members of a finite arithmetic progression is called an arithmetic series. Write a program in C++ to calculate the sum of the series (1*1) + (2*2) + (3*3) + (4*4) + (5*5) + + (n*n). Alternatively – put the first few terms of your sequence into WolframAlpha. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. Next: Write a program in C++ to find the sum of series 1 - X^2/2! + X^4/4!- upto nth term. That's numberwang!. Our summation calculator can easily calculate the sum of any numbers you input. Number sequences test practice for aptitude tests and psychometric IQ tests. sin105 degrees B. the starting principal you'll need to achieve the payouts desired:. , x k , we can record the sum of these numbers in the following way:. Examples: SAS Statements. Lets say i have three sequences. Notice, that if x = 1 then, in the series, we are simply addi ng up an infinite number of 1's and of course the sum goes to infinity. Please help. An infinite series has an infinite number of terms. Beware: people often confuse the terms ‘sequence’ and ‘series’. An Efficient solution to solve the sum of geometric series where first term is a and common ration is r is by the formula :-sum of series = a(1 – r n)/(1 – r). Enter 2 numbers to add and press the = button to get the sum result. #a_n = (3/2)^n# Which means that #n#-th term is generates by raising #3/2# to the #n#-th power. Find the series. Sequences and summations CS 441 Discrete mathematics for CS M. [Note that the sequence of the sums S 1 (the sum of the first term), S 2 (the sum of the first two terms), S 3 (the sum of the first three terms),. That's numberwang!. This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. +N^2 in C Programming Language. In other words, if you keep adding together the terms of the sequence forever, you will get a finite value. Products Classroom Activities Graphing Calculator Scientific Calculator Four Function Calculator Matrix Calculator Test Practice Geometry Tool. In other words, how to take the value of a cell located in one worksheet and add it to the value of another cell located in another worksheet to come up with the total of the respective cells. Series calculator allows to calculate online the sum of the terms of the sequence whose index is between the lower and the upper bound. sum_arithmetic_series is more general than required. what could be the problem? // Program that uses a switch statement. If this happens, we say that this limit is the sum of the series. P Class 11 Engineering - Sum of n Terms of G. All you have to do is write the first term number in the first box, the second term number in the second box, third term number in the third box and the write value of n in the fourth box after that you just have to click on the Calculate button, your result will be visible. Enter the sequence, the start value and end value from sigma notation and get a numerical sum. To add floating point values with extended precision, see math. What is the difficulty level of this exercise? Easy Medium Hard. About Sum (Summation) Calculator. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Calculate the sum of first n squares or the sum of consecutive square numbers from n 1 2 to n 2 2. A series is a special type of sequence: a. Notice, that if x = 1 then, in the series, we are simply addi ng up an infinite number of 1's and of course the sum goes to infinity. Calculate sum elements of sequence: sum. Thanks for your help. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. Bob has a need to use the SUM function in a macro in order to find the sum of all the values in a column. The sequence of partial sums of a series sometimes tends to a real limit. var ([ddof]) Calculate unbiased window variance. 23) a 21 = −1. The formula for the n-th term of a quadratic sequence is explained here. For example, some series don't sum monotonically to a limit -- the sum will bounce around it, and the summation's state can end up oscillatory forever. 02 Line 10 Column 42 Pic zz9 From N Highlight. skipna bool, default True. Third, we square the sum of X (45 times itself = 2025) and divide it by N (number of scores). Free Summation Calculator. +N^2 in C Programming Language. An example of the sequence can be seen as follows:. Instructions: Use this step-by-step Geometric Series Calculator, to compute the sum of an infinite geometric series by providing the initial term \(a$$ and the constant ratio $$r$$. Our sum of series calculator or arithmetic series calculator is an online tool which you can find on Google. Although SUM is specified as taking a maximum of 30 arguments, Google Sheets supports an arbitrary number of arguments for this function. About this calculator. (for example 2 series in the picture) 1 Comment. Free Arithmetic Sequences calculator - Find indices, sums and common difference step-by-step This website uses cookies to ensure you get the best experience. The sum function can be used as a series calculator, to calculate the sequence of partial sums of a series. Set Number = 0. The sums we have looked at so far are finite sums with finite upper and lower limits. The Arithmetic series of finite number is the addition of numbers and the sequence that is generally followed include – (a, a + d, a + 2d, …. day, sum((random()*5)::integer) num FROM days -- left join other tables here to get counts, I'm using random group by days. #sum_(k=1)^nb=nb# So. For example, the sum given by, means to sum an infinite number of terms as, The value of an infinite sum may be ∞ (in this case the sum is infinite). Definition: Arithmetic sequence is a list of numbers where each number is equal to the previous number, plus a constant. Using the while loop to calculate sum : While Loop « Statement Control « Java Tutorial. To add floating point values with extended precision, see math. Series calculator allows to calculate online the sum of the terms of the sequence whose index is between the lower and the upper bound. I can''t seem to find one of those. Build your own widget. A harmonic sequence is a sequence of numbers whose reciprocals form an arithmetic sequence. Here is the complete Java program with sample outputs. An Efficient solution to solve the sum of geometric series where first term is a and common ration is r is by the formula :-sum of series = a(1 – r n)/(1 – r). Let, t n be the n th term of AP, then (n+1) th term of can be calculated as (n+1) th = t n + D where D is the common difference (n+1) th - t n The formula to calculate N th term t n = a + (n – 1)d; where, a is first term of AP and d is the common difference. If Sum > Limit, terminate the repitition, otherwise. In the sequence a sub 1 (first term), a sub 2 (second term), a sub 3 (third term)…. Sum uses the standard Wolfram Language iteration specification. The last number n is input by the user. Σ is the symbol used to denote sum. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. Increment Number by one. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. 63333333 I'm having difficulty estimating how far this partial sum is away from the total sum. The blog talks about variety of topics on Embedded System, 8085 microprocessor, 8051 microcontroller, ARM Architecture, C2000 Architecture, C28x, AVR and many many more. Instructions: Use this step-by-step Geometric Series Calculator, to compute the sum of an infinite geometric series by providing the initial term $$a$$ and the constant ratio $$r$$. Arithmetic Sequence. After learning so much about development in Python, I thought this article would be interesting for readers and to myself… This is about 5 different ways of calculating Fibonacci numbers in Python [sourcecode language=”python”] ## Example 1: Using looping technique def fib(n): a,b = 1,1 for i in range(n-1): a,b = b,a+b return a print … Continue reading 5 Ways of Fibonacci in Python →. The 'nth' term is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next. 2, ALPHA ([I], 1, 1 0) EXE, and you should get an answer of 9. A sequence is a numbered list of values, produced by the calculation of a formula. C Program to calculate sum of 5 subjects and find percentage [crayon-5f51beda86fcb863628413/] Output : [crayon-5f51beda86fd2634443803/]. This is a free javascript calculator. The Fibonnacci numbers are also known as the Fibonacci series. Sequence Calculator, Sequence Examples. Improve your math knowledge with free questions in "Find the sum of an arithmetic series" and thousands of other math skills. Finding a general expression for a partial sum by induction and then finding the limit of this partial sum is a perfectly valid technique. How to find the sum of a finite Arithmetic Series! s n = n(t 1 + t n)/2 To find the sum of a finite arithmetic series, you need to know three things. Without using loops, calculate the sum of the following series for the first n terms. You may want to review the basics of geometric sequences or finding formulas. Notice, that if x = 1 then, in the series, we are simply addi ng up an infinite number of 1's and of course the sum goes to infinity. A good sequence to start with is the Fibonacci sequence. Repeat the following: a. About this calculator. sum_arithmetic_series FIRST LAST SKIP NB. #sum_(k=1)^nk=(n(n+1))/2# and. Definition: Geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a constant. How do we apply these useful rules to this question? First, calculate the average of the first and. This formula shows that a constant factor in the summands can be taken out of the sum. I have made this small program to help me add numbers and find the sum, it runs but no output. 2) ∧ 2 × 0. LED series current limiting resistor calculator - useful when designing circuits with a single LED or series/parallel LED arrays - for both the common small-current (20mA) LEDs and the more expensive, high power LEDs with currents up to a few Amperes. #sum_(k=1)^n(ak+b)# where a and b are constants. Free Summation Calculator. If S n tends to a limit as n tends to infinity, the limit is called the sum to infinity of the series. To add floating point values with extended precision, see math. 0 / k if k % 2 else -1. Fibonacci sequence formula; Golden ratio convergence; Fibonacci sequence table; Fibonacci sequence calculator; C++ code of Fibonacci function; Fibonacci sequence formula. 78 degrees Fahrenheit. Click the link for more information. The n-th partial sum of a series is the sum of the ﬁrst n terms. asked by ryan on March 15, 2017; Algebra. White Rose Maths has prepared a series of Maths lessons online for Year 4, FREE videos and worksheets. This calculator computes n-th term and sum of geometric progression person_outline Timur schedule 2011-07-16 04:17:35 Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. We start with the general formula for an arithmetic sequence of $$n$$ terms and sum it from the first term ($$a$$) to the last term in the sequence ($$l$$):. Explain why or why not. This is a very versatile calculator that will output sequences and allow you to calculate the sum of a sequence between a starting item and an n-th term, as well as tell you the value of the n-th term of interest. Calculate the rolling quantile. Since an arithmetic sequence always has an unbounded long-term behavior, we are always restricted to adding a finite number of terms. For this reason, Sum evaluates to null instead of to zero for an empty sequence or for a sequence that contains only nulls. So 355 minus 289. If the above series converges, then the remainder R N = S - S N (where S is the exact sum of the infinite series and S N is the sum of the first N terms of the series) is bounded by 0< = R N <= (N. Limits capture the long-term behavior of a sequence and are thus very useful in bounding them. The sum to infinity of a geometric progression. This version provided Java with a functional aspect by introducing concepts such as functional interfaces, lambda expressions, streams, etc. Sum of 32-bit integer quantities is not computed by using 64-bit results, and overflow can occur for the LINQ to SQL translation of Sum. Besides finding the sum of a number sequence online, server finds the partial sum of a series online. NPV = Sum CF* ((1+2%)/(1+D))^N When N is infinite, after simplification, NPV = CF * 1 / (1 - r) where r = (1+2%)/(1+D)--A+ V. Solving mathematical problems online for free. Since there's two equally likely options, you'd expect a run to last for two flips. See full list on gigacalculator. Sum of sequence calculator. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big. Therefore, the equation could be 4n. A geometric sequence refers to a sequence wherein each of the numbers is the previous number multiplied by a constant value or the common ratio. Calculate the sum of special series or number of sequences. Sum of Arithmetic Sequence Formula. com Task : To find the sum of all the elements in a list. Leonhard Euler continued this study and in the process solved many. Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums. Such series appear in many areas of modern mathematics. In the previous section we started looking at writing down a power series representation of a function. Geometric Sequence. The free tool below will allow you to calculate the summation of an expression. To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson Geometric sequence. skipna bool, default True. This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Store the result in an array. Perform addition to find the sum of two or more fractions. 8 Given two terms in an arithmetic sequence find the recursive formula. This calculator will find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). Hi everyone, the picture below is the sum of a series that have infinity limit, and i dont know the suitable function to deal with the problem. A sequence is a list of numbers, geometric shapes or other objects, that follow a specific pattern. For this reason, Sum evaluates to null instead of to zero for an empty sequence or for a sequence that contains only nulls. \$1000000007\$ is a prime number, so division by \$2^L\$ modulo 1000000007 is a multiplication by a multiplicative inverse of 2 modulo 1000000007 (which is trivial to find) to the same power. Calculate the sum of the series ∑ n = 1 ∞ a n whose partial sums are given. However, policy-makers disagree on the merits of this approach. Denote this partial sum by S n. An example of the sequence can be seen as follows:. Java program to calculate the sum of GP series. Fourth, we recall the sum of the X 2 and subtract 240. Work out: a. Online adding calculator. Sum of the series 1^1 + 2^2 + 3^3 + …. You may see the formula written as: Sum, S n, of n terms of an arithmetic series. The amount it increases or decreases by is known as the common difference. day ) select day, num, sum(num) over. +n and prints the result on the compiler screen. Two consecutive numbers in this series are in a ' Golden Ratio '. Flowchart. com allows you to find the sum of a series online. The Sum (Summation) Calculator is used to calculate the total summation of any set of numbers. 63333333 I'm having difficulty estimating how far this partial sum is away from the total sum. Such an argument was given by Nicolas Oresme (1323 - 1382 A. So the second term common to both sequences is 1+28 = 29. When the limit of partial sums exists, it is called the value (or sum) of the series. Dick and I both used tricks. It is capable of computing sums over finite, infinite and parameterized sequences. First simplify the expression, X! receives great quick! x/x! = x/(x * (x-a million) * (x-2) *a million) hence x/x! = a million/(x-a million)! for x >= 2 and a million for x =a million Then note that that's the same to the enlargement of the Taylor series for e^x as a million + x + x^2/2! + x^3/3! +x^4/4! for x = a million. You can also use this arithmetic sequence calculator as an arithmetic series calculator. If you are struggling to understand what a geometric sequences is, don't fret! We will explain what this means in more simple terms later on and take a look at. Here s how Wolfram Alpha solved it. In an Arithmetic Sequence the difference between one term and the next is a constant. Now, how do we determine the average of a sequence of integers? Rule #2: The average of a sequence of integers is the average of the first and last terms Applying the rules to find the sum of the sequence. #sum_(k=1)^n(ak+b)=sum_(k=1)^nak+sum_(k=1)^nb# We can factor the #a#. You need to know both of these numbers in order to calculate the sum of the arithmetic sequence. 2) ∧ 2 × 0. MATH 225N Week 4 Probability Questions and answers – Chamberlain College of Nursing Week 4 Homework Questions Probability 1. Only this variable may occur in the sequence term. Here s how Wolfram Alpha solved it. What Is Arithmetic Sequence? is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. day ) select day, num, sum(num) over. Find the sum of first n terms of the series (i) 3 +33 +333 +. Free Arithmetic Sequences calculator - Find indices, sums and common difference step-by-step This website uses cookies to ensure you get the best experience. The sequence of these partial sums converges to also. For example, in the sequence 10, 15, 20, 25, 30. The denominator doesn't change. Calculate the rolling quantile. Thus the message becomes: Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1. What two things do you need to know to find the sum of an infinite geometric series? Find the sum of the infinite geometric series. Bytes are provided as two-character strings. To find sum of your series, you need to choose the series variable, lower and upper bounds and also input the expression for n-th term of the series. Sum of Three Consecutive Integers calculator. Class 11 Commerce - Sum of terms of G. 2) ∧ 2 × 0. Let’s explore how we do that. Calculate the sum of series 1^2+2^2+3^2+4^2++n^2 (n>0) using the both for and while loop structure. Class 11 Commerce - Sum of terms of G. #sum_(k=1)^nk=(n(n+1))/2# and. On a higher level, if we assess a succession of numbers, x 1 , x 2 , x 3 ,. 63333333 I'm having difficulty estimating how far this partial sum is away from the total sum. If the value of the sum (in the limiting sense) exists, then they say that the series. After learning so much about development in Python, I thought this article would be interesting for readers and to myself… This is about 5 different ways of calculating Fibonacci numbers in Python [sourcecode language=”python”] ## Example 1: Using looping technique def fib(n): a,b = 1,1 for i in range(n-1): a,b = b,a+b return a print … Continue reading 5 Ways of Fibonacci in Python →. Calculators with newer operating systems actually have a summation function, which can by reached by pressing MATH 0. n must be a positive integer. In addition, when the calculator fails to find series sum is the strong indication that this series is divergent (the calculator prints the message like "sum diverges"), so our calculator also indirectly helps to get information about series convergence. In this case, a number of columns equal to the number of elements of the sequence x[n] should be skipped in order to accommo- date the results of the multiplication. Essentially the total variability in your dataset is the same as before, but now you have two different models to consider. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Free Summation Calculator. Fibonacci calculator The tool calculates F(n) - Fibonacci value for the given number, as well as the previous 4 values, using those to display a visual representation. The full sequence of keypresses to evaluate the Right-Hand Sum is: OPTN F4 [CALC] F6 F3 [Σ(] (1 + ALPHA ([I] × 0. This lesson assumes that you know about geometric sequences, how to find the common ratio and how to find an explicit formula. Some sources neglect the initial 0, and instead beginning the sequence with the first two ones. A geometric sequence refers to a sequence wherein each of the numbers is the previous number multiplied by a constant value or the common ratio. is 1 and any term is equal to the sum of all the succeeding terms. Unfortunately, the answer is NO. First simplify the expression, X! receives great quick! x/x! = x/(x * (x-a million) * (x-2) *a million) hence x/x! = a million/(x-a million)! for x >= 2 and a million for x =a million Then note that that's the same to the enlargement of the Taylor series for e^x as a million + x + x^2/2! + x^3/3! +x^4/4! for x = a million. This relationship of examining a series forward and backward to determine the value of a series works for any arithmetic series. Each number in series is called as Fibonacci number.


Insertion Sort on Small Arrays in Merge Sort

Although merge sort runs in $$\Theta(n \lg n)$$ worst-case time and insertion sort runs in $$\Theta(n^2)$$ worst-case time, the constant factors in insertion sort can make it faster in practice for small problem sizes on many machines. Thus, it makes sense to coarsen the leaves of the recursion by using insertion sort within merge sort when subproblems become sufficiently small. Consider a modification to merge sort in which $$n/k$$ sublists of length $$k$$ are sorted using insertion sort and then merged using the standard merging mechanism, where $$k$$ is a value to be determined.

1. Show that insertion sort can sort the $$n/k$$ sublists, each of length $$k$$, in $$\Theta(n \lg (n/k))$$ worst-case time.
2. Show how to merge the sublists in $$\Theta(n \lg(n/k))$$ worst-case time.
3. Given that the modified algorithm runs in $$\Theta(nk + n \lg(n/k))$$ worst-case time, what is the largest value of k as a function of n for which the modified algorithm has the same running time as standard merge sort, in terms of $$\Theta$$ notation?
4. How should we choose $$k$$ in practice?

#### Sorting Sublists

For input of size $$k$$, insertion sort runs on $$\Theta(k^2)$$ worst-case time. So, worst-case time to sort $$n/k$$ sublists, each of length $$k$$, will be $$n/k \cdot \Theta(k^2) = \Theta(nk)$$

#### Merging Sublists

We have $$n$$ elements divided into $$n/k$$ sorted sublists each of length $$k$$. To merge these $$n/k$$ sorted sublists to get a single sorted list of length $$n$$, we have to take 2 sublists at a time and continue to merge them.

This will result in $$\lg (n/k)$$ steps (refer to Figure 2.5 in page 38 of the chapter text). And in every step, we are essentially going to compare $$n$$ elements. So the whole process will run at $$\Theta(n \lg (n/k))$$.

#### Largest Value of k

For the modified algorithm to have the same asymptotic running time as standard merge sort, $$\Theta(nk + n \lg(n/k))$$ must be same as $$\Theta(n \lg n)$$.

To satisfy this condition, $$k$$ cannot grow faster than $$\lg n$$ asymptotically, if it does then because of the $$nk$$ term, the algorithm will run at worse asymptotic time than $$\Theta(n \lg n)$$).

So, let’s assume, $$k = \Theta(\lg n)$$ and see if we can meet the criteria …

\begin {aligned} \Theta(nk + n \lg(n/k)) & = \Theta(nk + n \lg n - n \lg k) \\ & = \Theta(n \lg n + n \lg n - n \lg (\lg n)) \\ & = \Theta(2n \lg n - n \lg (\lg n)) ^\dagger \\ & = \Theta(n \lg n) \end {aligned}

$$^\dagger\lg (\lg n)$$ is very small compared to $$\lg n$$ for sufficiently larger values of $$n$$.

#### Practical Value of k

To determine a practical value for $$k$$, it has to be the largest input size for which insertion sort runs faster than merge sort. To get exact value, we need to calculate the exact running time expressions with the constant factors and use the method described in Exercise 1.2.2.


# Are these $3$ functions linearly independent or dependent?

Are the functions $f, g, h$ given below linearly independent? If they are not linearly independent, find a nontrivial solutions to the equations below:

$$f(x)=e^{2x}- \cos(9x), \quad g(x)=e^{2x}+ \cos(9x), \quad h(x)= \cos(9x)$$

My take so far is that, I know these functions are not linearly independent (meaning this is surely dependent) by Wronskian (as the Wronskian determinant isn't equal to zero), but I have no idea how to find nontrivial solutions to this question. I think I should have an answer as follows:

$$C_1(e^{2x}-\cos(9x)) + C_2(e^{2x}+\cos(9x)) + C_3\cos(9x) = 0 \qquad \text{ where } Cs \text{ are constant }$$

Could I get some help on finding those constants? I've tried $C_1$ as $-1$, $C_2$ as $1$ and $C_3$ as $0$ but that surely cannot be the answer.

• In Mathematica, Wronskian[{Exp[2 x] - Cos[9 x], Exp[2 x] + Cos[9 x], Cos[9 x]}, x] yields $0$, showing that the three functions are linearly dependent. – David G. Stork Feb 8 '16 at 18:19

Those functions are a red herring. ;-)
You basically have the vectors $$v-w,\quad v+w,\quad w$$ in some vector space. These vectors belong to the subspace spanned by $v$ and $w$, which has dimension at most $2$. So a set of three vectors is necessarily linearly dependent.
How to find a nonzero linear combination is easy: $$a(v-w)+b(v+w)+cw=0$$ gives $$(a+b)v+(-a+b+c)w=0$$ so we can choose $b=-a$ and so $c=2a$. If we take $a=1$, we obtain $b=-1$ and $c=2$.
You can take $C_1 = 1, C_2 = -1$ and $C_3 = -2$.
• I choose $C_1 = 1$ (without any reason). Then I wanted to cancel the two exponential so I had to choose $C_2 = -1$. Then I was left with $-2\cos(9x)$ therefore I had to pick $C_3=-2$. – Onil90 Feb 8 '16 at 18:10
$$\begin{bmatrix} f(x)\\ g(x)\\ h(x) \end{bmatrix}= \begin{bmatrix} 1 & -1\\ 1 & 1 \\ 0 & 1 \\ \end{bmatrix}\cdot \begin{bmatrix} -e^{2x}\\ \cos(9x) \end{bmatrix}$$


# Average number of comparisons in sorted insertion

The questions is pretty simple: Given a sorted array of N elements, what is the average number of comparisons made in order to add a new element (let's call it x) in its correct position?
I am using linear search for that task.

So far, I've tried to solve it this way:

# of Comparisons     Probability of A[i] > x (Not sure if correct)
1                         1/n
2                         1/n
3                         1/n
4                         1/n
...                        ...
n                         1/n

Hence, the expected value would be:

$$E[x] = \sum_{x=1}^{n}x \cdot Pr(X=x)$$ $$E[x] = 1 \cdot Pr(X=1) + 2 \cdot Pr(X=2) + 3 \cdot Pr(X=3) + ... + n \cdot Pr(X=n)$$

Using $$Pr(X = n) = \frac{1}{n}$$ for all n,

$$E[x] = \frac{1}{n} \cdot \sum^{n}_{i=1}{i}$$

and finally $$E[x] = \frac{n+1}{2}$$

Still, I'm not sure if this is the correct way to solve it, since I'm not sure about my 1/n assumption.

Well, that would strongly depend on the algorithm that you use to find out where to insert the new element, and on the distribution of new elements.

Assuming that you do a linear search, and the new element could go to any position with the same probability, your result is close, but not quite exact. The new element can go into one of (n + 1) positions, not n. There may be up to n comparisons needed; n comparisons are needed both if the new element goes into the first array position, and if it goes into the second array position. So the result is

(1 + 2 + 3 + ... + (n-1) + n + n) / (n + 1) =

= (n (n+1) / 2 + n) / (n + 1)

= n (n + 3) / 2 / (n + 1)

= (n + 2 - 2 / (n + 1)) / 2 ≈ (n + 2) / 2,

which is just a tiny bit more than your answer (n + 1) / 2.

• Thanks a lot @gnasher729 ! I completely forgot to mention that I'd perform a linear search in order to find the position to insert the new number! – woz Sep 1 '17 at 23:29
• I don't understand what you mean by "n comparisons are needed both if the new element goes into the first array position, and if it goes into the second array position". Why do we need $n$ comparisons in order to insert $x$ into the first position? Assuming I compare by $x < A[i]$, a is single comparison $x < A[0]$ is enough to insert x into the first position. – fade2black Sep 1 '17 at 23:48
• @fade2black: Because you likely start comparing at the end of the array, which allows you to compare and move one element in the same iteration of the loop. If you start comparing at the beginning, then n comparisons are needed both if the new element is inserted just before, or just after the last element. – gnasher729 Sep 2 '17 at 22:38


# Show which of $6-2\sqrt{3}$ and $3\sqrt{2}-2$ is greater without using calculator

How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)

Look simple but I have tried many ways and fail miserably. Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$. Taking the first minus the second in order to see the result positive or negative get me no where (perhaps I am too dense to see through).

• sorry, multiply both sides by $6+2\sqrt{3}$ and use $(a-b)(a+b)=a^2-b^2$ – luka5z Jun 10 '17 at 19:00
• \begin{eqnarray*} \sqrt{a}+ \sqrt{b}= \sqrt{a+b+2 \sqrt{ab}} \end{eqnarray*} might be helpful. – Donald Splutterwit Jun 10 '17 at 19:01
• @Learner132 A lot of the solutions here add or subtract from both sides then square then compare to infer about the direction of the original identity. This is only guaranteed to work if both values are positive before squaring. E.g. consider the counterexample: $2.5>1 \rightarrow 2.5-2>1-2 \rightarrow 0.5 > -1 \rightarrow 0.25 > 1$. – Ian Miller Jun 11 '17 at 11:49
• I would just guesstimate. $\sqrt 3 \approx 1.75$, $\sqrt 2 \approx 1.4$, so $6 - 2 \sqrt 3 \approx 2.5$ and $-2 + 3 \sqrt 2 \approx 2.2$. – Robert Soupe Jun 15 '17 at 0:39

$6-2\sqrt 3 \gtrless 3\sqrt 2-2$

Rearrange: $8 \gtrless 3\sqrt 2 + 2\sqrt 3$

Square: $64 \gtrless 30+12\sqrt 6$

Rearrange: $34 \gtrless 12\sqrt 6$

Square: $1156 \gtrless 864$

• You might simplify by $2$ before squaring… – Bernard Jun 10 '17 at 19:11
• Thank you, good answer, but I doubt if math teacher is okay with this (> or <) symbol – Learner132 Jun 10 '17 at 19:29
• What's important about any symbol is its mathematical meaning. If you can the mathematical meaning of this symbol to your math teacher, I'll bet he/she is open to its usage. – Lee Mosher Jun 10 '17 at 19:57
• You should probably mention that the squaring steps are valid (both forward and backward) since the values on both sides are clearly positive. – Rory Daulton Jun 11 '17 at 0:33
• Yes, ordinarily you would start at the last step with a definite operator there (> in this case) and then work towards the inequality given to you. However, to find the necessary steps, it is often easier to work from what you’re given to something known, i.e., backwards. – BallpointBen Jun 11 '17 at 2:47

We have $\sqrt{3}\leq 1.8$ so $6-2\sqrt{3}\geq 2.4$, whereas $\sqrt{2}\leq 1.42$ so $3\sqrt{2}-2\leq 2.26$.

• I love your answer, I'll try to see if math teacher is okay with estimation – Learner132 Jun 10 '17 at 19:31
• @Learner132: Note that the estimates are easy to prove by squaring, if necessary. – user14972 Jun 10 '17 at 22:06

$$6-2√3 \sim 3√2-2\\ 8 \sim 3√2 +2√3 \\ 64 \sim 30+12√6\\ 34 \sim 12√6\\ 17 \sim 6√6\\ 289 \sim 36 \cdot 6\\ 289 > 216$$

• Thanks, quite understandable, but how I can explain ~ symbol to math teacher in this? – Learner132 Jun 10 '17 at 19:27
• This $\sim$ simbol is a placeholder for an undefined relationship, you can use whatever other symbol you prefer, while you dont multiply by -1 – Brethlosze Jun 10 '17 at 19:29
• I remember using that during school first courses without trouble – Brethlosze Jun 10 '17 at 19:29
• I understand you. Well, I hope math teacher is okay with it although it is not covered in the lecture. – Learner132 Jun 10 '17 at 19:40
• Teachers know how this work and what you thought... rewritting everything is not needed. – Brethlosze Jun 11 '17 at 6:17

Using simple continued fractions for $\sqrt {12}$ and $\sqrt {18}.$ Worth learning the general technique...Method described by Prof. Lubin at Continued fraction of $\sqrt{67} - 4$

$$\sqrt { 12} = 3 + \frac{ \sqrt {12} - 3 }{ 1 }$$ $$\frac{ 1 }{ \sqrt {12} - 3 } = \frac{ \sqrt {12} + 3 }{3 } = 2 + \frac{ \sqrt {12} - 3 }{3 }$$ $$\frac{ 3 }{ \sqrt {12} - 3 } = \frac{ \sqrt {12} + 3 }{1 } = 6 + \frac{ \sqrt {12} - 3 }{1 }$$

Simple continued fraction tableau:
$$\begin{array}{cccccccccc} & & 3 & & 2 & & 6 & \\ \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 3 }{ 1 } & & \frac{ 7 }{ 2 } \\ \\ & 1 & & -3 & & 1 \end{array}$$

$$\begin{array}{cccc} \frac{ 1 }{ 0 } & 1^2 - 12 \cdot 0^2 = 1 & \mbox{digit} & 3 \\ \frac{ 3 }{ 1 } & 3^2 - 12 \cdot 1^2 = -3 & \mbox{digit} & 2 \\ \frac{ 7 }{ 2 } & 7^2 - 12 \cdot 2^2 = 1 & \mbox{digit} & 6 \\ \end{array}$$

Continued fraction convergents alternate above and below the irrational number, we get $$\frac{ 3 }{ 1 } < \sqrt {12} < \frac{ 7 }{ 2 }$$ Your first number was $6 - \sqrt {12},$ $$3 > 6 - \sqrt {12} > \frac{ 5 }{ 2 }$$ $$\frac{ 5 }{ 2 } < 6 - \sqrt {12} < 3$$

Next 18......................========================================

$$\sqrt { 18} = 4 + \frac{ \sqrt {18} - 4 }{ 1 }$$ $$\frac{ 1 }{ \sqrt {18} - 4 } = \frac{ \sqrt {18} + 4 }{2 } = 4 + \frac{ \sqrt {18} - 4 }{2 }$$ $$\frac{ 2 }{ \sqrt {18} - 4 } = \frac{ \sqrt {18} + 4 }{1 } = 8 + \frac{ \sqrt {18} - 4 }{1 }$$

Simple continued fraction tableau:
$$\begin{array}{cccccccccc} & & 4 & & 4 & & 8 & \\ \\ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 4 }{ 1 } & & \frac{ 17 }{ 4 } \\ \\ & 1 & & -2 & & 1 \end{array}$$

$$\begin{array}{cccc} \frac{ 1 }{ 0 } & 1^2 - 18 \cdot 0^2 = 1 & \mbox{digit} & 4 \\ \frac{ 4 }{ 1 } & 4^2 - 18 \cdot 1^2 = -2 & \mbox{digit} & 4 \\ \frac{ 17 }{ 4 } & 17^2 - 18 \cdot 4^2 = 1 & \mbox{digit} & 8 \\ \end{array}$$

This time the number is $\sqrt {18} - 2.$

It is enough to use $$2 < \sqrt {18} - 2 < \frac{9}{4}$$

$$\color{red}{ 2 < \sqrt {18} - 2 < \frac{9}{4} < \frac{ 5 }{ 2 } < 6 - \sqrt {12} < 3 }$$

• Thank you, but I'll study it one day – Learner132 Jun 10 '17 at 19:51

Define

$a=6-2\sqrt 3>0$

$b=3\sqrt 2-2>0$

$a-b = 8 - (2\sqrt 3 + 3\sqrt 2)$

$(2\sqrt 3 + 3\sqrt 2)^2 = 30+12\sqrt 6 = 6×(5+2\sqrt 6) < 60 < 64$ because $6=2×3 < (5/2)^2$

$a-b > 8-8=0, a>b$

• Thank you, it took me a while to understand 2×3<(5/2)^2 but it is easier to explain to math teacher. – Learner132 Jun 10 '17 at 19:45

Here's yet another way, for those who aren't comfortable with the $\gtrless$ or $\sim$ notation.

We can use crude rational approximations to $\sqrt 2$ and $\sqrt 3$.

\begin{align} \left(\frac{3}{2}\right)^2 = \frac{9}{4} & \gt 2\\ \frac{3}{2} & \gt \sqrt 2\\ \frac{9}{2} & \gt 3\sqrt 2 \end{align}

And \begin{align} \left(\frac{7}{4}\right)^2 = \frac{49}{16} & \gt 3\\ \frac{7}{4} & \gt \sqrt 3\\ \frac{7}{2} & \gt 2\sqrt 3 \end{align}

Adding those two approximations, we get \begin{align} \frac{9}{2} + \frac{7}{2} = 8 & \gt 3\sqrt 2 + 2\sqrt 3\\ 6 + 2 & \gt 3\sqrt 2 + 2\sqrt 3\\ 6 - 2\sqrt 3 & \gt 3\sqrt 2 - 2 \end{align}

I'll use >=< to represent the unknown comparison.

$6-2\sqrt{3} >=< 3\sqrt{2}-2$

Lets start by adding two to both sides to reduce the number of numbers. This doesn't change the comparison result.

$8-2\sqrt{3} >=< 3\sqrt{2}$

Both sides are clearly positive ( $2\sqrt{3} < 6$ ) so we can square both sides without changing the comparison result.

In a more maginal case where we were unsure if the left hand side was positive we could have compared the two terms in the left hand side by squaring both of them and hence determined whether the left hand side was positive or negative.

$64 -32\sqrt{3} + 12 >=< 18$

Now lets collect terms.

$60 >=< 32\sqrt{3}$

Divide by four.

$15 >=< 8\sqrt{3}$

Square again.

$225 >=< 64 \times 3$

$225 > 192$

Therefore

$6-2\sqrt{3} > 3\sqrt{2}-2$

There is still some hope in taking the first minus the second in this case: $$6-2\sqrt{3} - (3\sqrt{2}-2) = 8 - (2\sqrt{3} + 3\sqrt{2})$$

So now the question boils down to if the expression with the square root exceeds $8$. We know that $8^{2} = 64$ and: $$(2\sqrt{3}+3\sqrt{2})^{2}=4*3+2(2\sqrt{3})(3\sqrt{2})+9*2 =30+12\sqrt{3}\sqrt{2}$$

Conversely: $$8^{2} = 64 = 28+36=28+6*6 = 28+6*\sqrt{6}\sqrt{6}$$

Subtracting them yields: $$28+6*\sqrt{6}\sqrt{6}-30+12\sqrt{3}\sqrt{2} = -2+6*\sqrt{6}\sqrt{6}-12\sqrt{3}\sqrt{2}$$

Thus this is only positive if the square root terms are positive and exceed 2. Comparing the square root terms: $$6*\sqrt{6}\sqrt{6}-12\sqrt{3}\sqrt{2} = 6*\sqrt{2}\sqrt{3}(\frac{\sqrt{2}}{\sqrt{2}})\sqrt{6}-12\sqrt{6}=12\frac{\sqrt{3}}{\sqrt{2}}\sqrt{6}-12\sqrt{6}=12\sqrt{6}(\frac{\sqrt{3}}{\sqrt{2}}-1)$$

We conclude that the square root term is positive since $\sqrt{3}>\sqrt{2}$ . But is it greater than $-2$? Or rather, how much do we need to multiply to the expression $\frac{\sqrt{3}}{\sqrt{2}}-1$ for it to be greater than $2$? $$n*(\frac{\sqrt{3}}{\sqrt{2}}-1)>2$$ $$n>\frac{2}{\frac{\sqrt{3}}{\sqrt{2}}-1}*\frac{\frac{\sqrt{3}}{\sqrt{2}}+1}{\frac{\sqrt{3}}{\sqrt{2}}+1}\approx\frac{2*(1.5+1)}{0.5} = 10$$ So the factor in front of the term $\frac{\sqrt{3}}{\sqrt{2}}-1$ should be at least 10. But since $12>10$ and $\sqrt{6}>1$, we conclude that $12\sqrt{6}>10$. Thus, substituting back to the original equation: $$-2+6*\sqrt{6}\sqrt{6}-12\sqrt{3}\sqrt{2} = -2+12\sqrt{6}(\frac{\sqrt{3}}{\sqrt{2}}-1) > 0$$ And so we conclude that: $$64>(2\sqrt{3}+3\sqrt{2})^{2}$$ and so, for positive root, $$8>(2\sqrt{3}+3\sqrt{2})$$ and $$8 - (2\sqrt{3} + 3\sqrt{2}) > 0$$

You may use other approaches. Notice that I split $8^{2} = 28+36$. Equally viable is to split $8^{2} = 30+34$, and you might have to use a similar (Squaring than rooting) trick to show that $34 > 12\sqrt{6}$, and then finally conclude that $8>(2\sqrt{3} + 3\sqrt{2})$. This alternative approach should be able to give you a smaller threshold compared to the approximation I made midway.

$6-2√3 \approx2(1.732) = 6 - 3.464 = 2.536$ $3√2-2 \approx 3(1.414) - 2 = 4.242 - 2 = 2.242$

This implies that $\dfrac 52$ is between the two quantities:

\begin{align} \dfrac{49}{4} &> 12 \\ \dfrac 72 &> 2\sqrt 3 \\ \dfrac{12}{2} &> 2\sqrt 3 + \dfrac 52 \\ 6 - 2\sqrt 3 &> \dfrac 52 \end{align}

and

\begin{align} \dfrac{81}{4} &> 18 \\ \dfrac 92 &> 3\sqrt 2 \\ \dfrac 52 &> 3\sqrt 2 - 2 \end{align}

It follows that $6-2√3 > 3\sqrt 2 - 2$.

• see this answer same concept but better (less digits, can be easily verified by hand, using lower and upper bounds instead of $\approx$)) and 12 hours earlier than your answer – miracle173 Jun 11 '17 at 14:16
• @miracle173 What you saw was only a part of what I intended to do. I needed some more time to think about the problem so I thought I had deleted everything. Obviously, I hadn't. What you see above is what I intended to do. – steven gregory Jun 11 '17 at 19:57


# Find the average velocity

If a ball is thrown in the air with a velocity $34$ ft/s, its height in feet $t$ seconds later is given by

$$y = 34 t − 16 t^2 .$$

Find the average velocity for the time period beginning when $t = 2$ and lasting $0.5$ second, $0.1$ second, $0.05$ second, $0.01$ second and estimate the instantaneous velocity when $t = 2$.

I tried to solve by doing the following:

$y=34(2)-16(2)^2$

$y=68-64$

$y=a$

$4/0.5=8$ft/s

but I was told that answer is incorrect.

What did I do wrong?

• This is a limit approximation. If we say $y=f(x)$, then we can write that the first average is ${f(2.5)-f(2)\over 0.5}={-15-4\over 0.5}=-38$. – abiessu Sep 18 '13 at 21:50
• How did you get -15 for f(2.5) and 4 for f(2) ? – Grey Sep 18 '13 at 22:22
• I used $f(2.5) = 34(2.5)-16(2.5)^2 = 85-100=-15$ and $f(2)=34(2)-16(2)^2=68-64=4$. I saw the $62$ in the second term of your original post before it was edited and that $16\cdot 4$ is not $62$... – abiessu Sep 18 '13 at 22:26
• Thanks. And how to get get the instantaneous velocity when t = 2 ? – Grey Sep 18 '13 at 22:40
• That is the point of the exercise, first calculate $f(2.5)-f(2)\over 0.5$, then $f(2.1)-f(2)\over .1$, and so on, and use this series of values to estimate the instantaneous velocity at $t=2$. – abiessu Sep 18 '13 at 22:42

You are being asked for the average velocity over the time span $2$ to $2.5$ seconds, among others. To do that one, you need $y(2.5)$ and $y(2)$ Then the average velocity in that span is $\frac {y(2.5)-y(2)}{2.5-2}$

There is enough information here to complete the following process:

Given $y(t)=34t-16t^2$, we have:

$$y(2)=34\cdot 2-16\cdot 4=68-64=4$$

$$y(2.5)=34\cdot 2.5-16\cdot 2.5^2=85-100=-15$$

$$y(2.1)=34\cdot 2.1-16\cdot 2.1^2=71.4-70.56=0.84$$

$$y(2.05)=2.46$$ (using calculator)

$$y(2.01)=3.6984$$ (using calculator)

Now, the remaining task is to calculate $-19\over 0.5$, $-3.16\over 0.1$, $-1.54\over 0.05$, and $-0.3016\over 0.01$ and see if these numbers are approaching a particular number. I got each of these fractions by writing $y(2+t)-y(2)\over t$ for each of the values $t\in \{0.5,0.1,0.05,0.01\}$.


Mathematics behind this card trick

1. Suppose I have $21$ playing cards. I distribute them in $3$ columns and tell you to choose mentally a card. Then just indicate in which column the card is.

2. I pick up one of the columns which doesn't contain your card, then the column which contains your card then the remaining column.

3. Now I deal the cards in $3$ columns again, starting from the left to the right, and repeating the process until there are no cards left in my hand. I ask you to indicate to me in which column your card is.

4. I repeat step 2 then 3.

5. I repeat step 2.

Now by counting either way in the deck of $21$ cards, your card will be the 11th card.

I've tried using modulo to understand the problem but I'm stuck doing integer divisions. So does anyone have a simpler way to explain the trick.

Also could anyone explain why it works only for odd number of cards in each column and why it requires additional steps for more cards. e.g for $17 \cdot 3 = 51$ requires an additional step compared with $21$ cards?

EDIT: I forgot to add that the number of cards you have to count for the final step equals

$$1.5\text{ times the number of cards in each column} + 0.5$$

• If you do not have an odd number of cards in total, there is not a middle card Dec 12 '16 at 13:25
• If you have $n$ cards and $3$ columns, you need $\log_3(n)$ steps (rounded up) to distinguish them Dec 12 '16 at 13:26
• This related trick (and its explanation) might provide some insight Dec 12 '16 at 13:29
• I first learnt this card trick when I was 14. After learning some calculus, I was pleasant to realise that the fact that the card will eventually end up at the middle position is basically a consequence of the squeezing principle. Dec 13 '16 at 9:13
• @user1551 That seems like an overapplication of the squeezing principle.
– jwg
Dec 13 '16 at 10:30

Suppose that, when you first lay the cards on the table, the card I choose is at position $x$ in its column. You don't know $x$, but you know that $1\leq x\leq 7$.

Now, when you pick up the cards, my card will be at position $7+x$ in the full stack. The second time you lay the cards on the table, my card will appear at position $p_1=\lceil\frac{7+x}{3}\rceil$ in its column.

The second time you pick up the cards, my card will be at position $7+p_1$ in the full stack. The third time you lay the cards on the table, my card will appear at position $p_2=\lceil\frac{7+p_1}{3}\rceil$ in its column.

Finally, when you pick up the cards for the third time, my card will be at position $7+p_2$ in the full stack. Putting this all together, my card will be at position $$7+p_2=7+\left\lceil\frac{7+\lceil\frac{7+x}{3}\rceil}{3}\right\rceil$$ in the full stack. The trick is that this is equal to $11$ for all $x$ in the range $1\leq x\leq 7$.

For a proof of this last statement, as jpmc26 mentions, one can apply the identities $\lceil\frac{m+\lceil x\rceil}{n}\rceil=\lceil\frac{m+x}{n}\rceil$ and $\lceil n+x\rceil = n+\lceil x\rceil$ (for real $x$, integer $m$, and positive integer $n$) to show that $$7+\left\lceil\frac{7+\lceil\frac{7+x}{3}\rceil}{3}\right\rceil = 7+\left\lceil\frac{7+\frac{7+x}{3}}{3}\right\rceil = 7 + \left\lceil 3 + \frac{x+1}{9}\right\rceil = 10 + \left\lceil\frac{x+1}{9}\right\rceil \enspace,$$ which is clearly equal to $11$ for $1\leq x\leq 7$.

• You can use the fact that $\left \lceil \frac{7+\lceil y \rceil}{3} \right \rceil = \left \lceil \frac{7+ y }{3} \right \rceil$ (see here) to show that the last expression is equal to $7 + \left \lceil 3 + \frac{x + 1}{9} \right \rceil = 10 + \left \lceil \frac{x + 1}{9} \right \rceil$, which makes the fact it's always 11 much more obvious. Dec 12 '16 at 23:14
• Good point. I just checked that it holds for x=1 and for x=7, therefore it holds for x in between due to monotonicity. Should I edit the answer to include your remark? Dec 13 '16 at 8:17
• @EvangelosBampas: Yes! Dec 13 '16 at 12:47
• @EvangelosBampas Absolutely. (Thanks!) Comments are considered ephemeral/transient, and their primary purpose is for the improvement of a post. You can generally assume that useful content or clarification posted in them warrants an edit or an additional post. (In this case, I only provided some comparatively minor additions that just build on your answer and make the conclusion easier to see, so a separate post wouldn't be able to stand on its own.) Dec 13 '16 at 14:13

You can find a full explanation of this trick and related ones at

Gergonne’s Card Trick, Positional Notation, and Radix Sort
(Mathematics Magazine, February, 2010)

http://www.maa.org/sites/default/files/Bolker-MMz-201053228.pdf

The classic trick uses $27 = 3^3$ cards. Your $21$ card version is discussed on page 48.

• This answer comes dangerously close to "link only." Dec 12 '16 at 22:51
• I can see a total of 4 lines floating freely besides the link provided, which will definitly assure that the answer is too far from being called a link only if anything. Dec 13 '16 at 1:01
• @jpmc26 I don't like link only answers either, but the relevant content in the link is too long to post - I'd essentially be pasting in most of the paper. The link is to the journal, so stable. Dec 13 '16 at 2:06

After one deal-and-gather phase, your card is in the middle third of the deck, right?

When you deal out the cards again, where does that middle third end up? In the middle third (looking top-to-bottom) of the tableau.

To see this: on the first deal, pick a card, say, the 3 of hearts. Replace everything else in its column with red cards, and everything else in the other two columns with black cards. Gather and re-deal. You'll see a red "band" in the middle (top to bottom) of the new tableau.

Now you identify your card in that middle band. When you gather up the other cards, it'll once again be in the middle third of the deck, because there's one entire column-worth of cards in front of it, and one entire column-worth of cards behind it.

But you can say more than that: suppose your card was in the first column. Well, then, it was in the middle third of the first column, wasn't it? So if each column has $k$ cards, you've got $k$ cards in front of it (from one of the other piles), and another k/3 cards (from its own pile) in front of it, and the same for cards behind it. So now it's in the middle 1/9th of the pile.

And the next time it'll be in the middle 1/27 of the pile. And so on.

Now that analysis isn't QUITE right, because $k$ might not be divisible by 3. So instead of having k + k/3 cards in front of it after two deals, you have k + floor(k/3). For instance, if $k = 8$, then you'd have $8 + floor(8/3) = 8 + floor(2.66...) = 8 + 2 = 10$ cards in front of it. But aside from this minor glitch, what you get is the following:

after $p$ "passes" on a deck of $N$ cards, your card is in the middle (roughly) $(N/3^p)$ cards of the deck. When $3^p > N$, this means that your card is the middle card of the deck.

• +1, Definitely makes more sense to me than the algebra above. Dec 13 '16 at 1:00

Let's go step by step. After steps i) and ii) we already know that the chose card is in position $8,9,10,11,12,13,14$ in the pile. Then after dealing all the cards again in the given manner we know that the cards that were in the positions $8,9,10,11,12,13,14$ before dealing are now in places $3,4,5$ (actually it depends on the pile, but we can be sure that each of the mentioned cards is now at a position $3,4$ or $5$ in one of the three piles. So eventually after the second step we narrowed the choice to three cards.

After putting the cards in one pile again then we know that the chosen card will be in position $10,11,12$. And in the last deal obviously each of the cards will be in dealt in different pile, so we should be able to find out which one is the card for sure. But during the last deal the cards in positions $10,11,12$ are all in position $4$ in each of the new piles, hence after collecting all the cards, the chosen card will be in position $11$.

Eventually as you can see with each dealing we lower the number of possibilities from $n$ to $\lfloor\frac{n-1}{3}\rfloor + 1$.