# If $12$ distinct points are placed on a circle and all the chords connecting these points are drawn, at how many points do the chords intersect?

If $$12$$ distinct points are placed on the circumference of a circle and all the chords connecting these points are drawn, at how many points do the chords intersect? Assume that no three chords intersect at the same point.

A) $$12\choose2$$

B) $$12\choose4$$

C) $$2^{12}$$

D) $$\frac{12!}2$$

I tried drawing a circle and tried to find a pattern but couldn't succeed. for 12 points I found the answer to be $$(1+2+....+9)+(1+2+3+.....+8)+....+(1)$$ and the result multiplied by $$2$$. But I'm getting $$296$$ which is in none of the options. Can anyone help?

• Try it for fewer points than $12$ and see if you can spot a pattern. – saulspatz Dec 12 '18 at 15:08
• The numbers in (A)-(D) are all usually associated with counting certain (possibly ordered) subsets of a 12-element set. Can you tell what type of subsets these numbers count? Which type of subset corresponds to a single intersection point? – Mees de Vries Dec 12 '18 at 15:10
• @saulspatz for 12 points I found the answer to be (1+2+....+9)+(1+2+3+.....+8)+....+(1) and the result multiplied by 2. But I'm getting 296 which is in none of the options – Ayaz S Imran Dec 12 '18 at 15:14
• @MeesdeVries (A) is the number of intersection of 12 points – Ayaz S Imran Dec 12 '18 at 15:16
• You should add your result, and the method you used to obtain it, to the body of the question. (Don't make another comment. Edit the question.) Then we'll be able to tell you where you've gone wrong. – saulspatz Dec 12 '18 at 15:16

If you select any $$4$$ distinct points on the circle, you'd have one distinct point of intersection. This'll give you a nice little formula of selecting $$4$$ points out of $$n$$.

$$N={n\choose4}={12\choose4}=495$$

Let's follow the suggestion of @saulspatz by considering the problem for fewer points. Consider the diagram below.

The points on each circle have been chosen in such a way that no three chords intersect at the same point. Under these conditions, we can see by inspection that

$$\begin{array}{c c} \text{number of points} & \text{number of intersections}\\ \hline 4 & 1\\ 5 & 5\\ 6 & 15 \end{array}$$

This should suggest a formula for the number of intersections when we have $$n$$ points and no three chords intersect in the same point.

A chord is determined by two points of the circle. Two intersecting chords are determined by four points of the circle since the only way the chords can intersect is if we connect both pairs of nonadjacent points.


# Probability of a point in a square being 0.5 from perimeter

1. Feb 10, 2015

### Nathanael

1. The problem statement, all variables and given/known data
Consider a square of side length 1. Two points are chosen independently at random such that one is on the perimeter while the other is inside the square. Find the probability that the straight-line distance between the two points is at most 0.5.

2. The attempt at a solution
I assumed that I can just treat the point on the perimeter as if it's chosen at random along half of 1 side (from a corner to a midpoint) because no matter where the random point on the perimeter is chosen, the square can be reoriented (in our imagination) so that it lies on the same line segment (from a corner to the midpoint of a side). I feel I am justified in doing this because reorienting the square should not change the randomness of the point inside the square (because the point in the square is independently random, and also because all possible ways to reorient the square should be equally likely). Is this where I went wrong?

Let me proceed with this assumption.
Call the distance from the corner y. From the above assumption, y ranges from [0, 0.5]
Take a look at my drawing of the situation:

I think dy/0.5 = 2dy would be the chance of the random point "being" y (or being within dy of y, or however you should think of it)

So I think the solution to the problem should be $\int\limits_0^{0.5} 2A(y)dy$

And finally, $A(y)=\int\limits_0^{y+0.5} \sqrt{0.25-(y-x)^2}dx$

This gives me an answer of π/8-1/12 ≈ 0.31 which seems reasonable, but is apparently
incorrect.

(It seems reasonable because at most A(y)=A(1/2)=pi/8≈0.4 and at least A(y)=A(0)=pi/16≈0.2 so the answer should be somewhere between 0.2 and 0.4)

I'm not sure what I did wrong.

2. Feb 10, 2015

### Bystander

Your sketch shows point "y" at one extreme, however, it can be anywhere from the corner to halfway "up." My take, I guarantee nothing.

3. Feb 10, 2015

### RUber

What is the probability that a point would fall into the quarter circle if the point on the perimeter were at the corner?
You seem to have worked out the half-circle problem for when the point falls in the middle.
The relative probability will be $\int \int d(x,y) p(x) dx dy$

4. Feb 10, 2015

### Ray Vickson

I get your answer as well; why do you think it is wrong?

5. Feb 10, 2015

### BvU

Hello Nate/Richard :)

Did what you did, found what you found ${\pi\over 8} - {1\over 12}$.

Corroborated with a rude numerical simulation (16000 x 3 excel numbers :) , 12 times; found 0.30905 +/- 0.00071)

So here's a second one who agrees with you.
What tells you it's wrong ?

6. Feb 10, 2015

### Nathanael

Oh, wow... The problem asked, what is the probability that the distance will be at least 0.5... arghuhghgrumbles... 13/12 - pi/8

I am very sorry for wasting all your time!

7. Feb 10, 2015

### Ray Vickson

No time wasted at all. Computing 1 - P(d < .05) is by far the easiest way to compute P(d >= 0.5). This is clear at once if you refer back to your diagram.

8. Feb 10, 2015

### BvU

Second Ray. And it was an interesting exercise ! So thumbs up ! :)


nthRow(int N), Grinding HackerRank/Leetcode is Not Enough, A graphical introduction to dynamic programming, Practicing Code Interviews is like Studying for the Exam, 50 Data Science Interview Questions I was asked in the past two years. In this post, I have presented 2 different source codes in C program for Pascal’s triangle, one utilizing function and the other without using function. More rows of Pascal’s triangle are listed on the ﬁnal page of this article. around the world. For a more general result, see Lucas’ Theorem. The first and last terms in each row are 1 since the only term immediately above them is always a 1. / (i+1)! $${n \choose k}= {n-1 \choose k-1}+ {n-1 \choose k}$$ Suppose true for up to nth row. Using this we can find nth row of Pascal’s triangle. Complexity analysis:Time Complexity : O(n)Space Complexity : O(n), C(n, i) = n! Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. Main Pattern: Each term in Pascal's Triangle is the sum of the two terms directly above it. See all questions in Pascal's Triangle and Binomial Expansion. We often number the rows starting with row 0. To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. The 1st row is 1 1, so 1+1 = 2^1. Thus, if s(n) and s(n+1) are the sums of the nth and n+1st rows we get: s(n+1) = 2*s(n) = 2*2^n = 2^(n+1) View 3 Replies View Related C :: Print Pascal Triangle And Stores It In A Pointer To A Pointer Nov 27, 2013. (n-i-1)! C(n, i+1) / C(n, i) = i! For an alternative proof that does not use the binomial theorem or modular arithmetic, see the reference. Naive Approach:Each element of nth row in pascal’s triangle can be represented as: nCi, where i is the ith element in the row. The elements of the following rows and columns can be found using the formula given below. Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. Just to clarify there are two questions that need to be answered: 1)Explain why this happens, in terms of the way the triangle is formed. For example, the numbers in row 4 are 1, 4, 6, 4, and 1 and 11^4 is equal to 14,641. Here are some of the ways this can be done: Binomial Theorem. How do I use Pascal's triangle to expand #(3a + b)^4#? Each number is the numbers directly above it added together. ((n-1)!)/((n-1)!0!) QED. As we know the Pascal's triangle can be created as follows − In the top row, there is an array of 1. It's generally nicer to deal with the #(n+1)#th row, which is: #((n),(0))# #((n),(1))# #((n),(2))# ... #((n),(n))#, #(n!)/(0!n! How do I use Pascal's triangle to expand the binomial #(d-3)^6#? How do I use Pascal's triangle to expand the binomial #(a-b)^6#? Suppose we have a number n, we have to find the nth (0-indexed) row of Pascal's triangle. 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08 Jan 2021

## nth row of pascal's triangle

So a simple solution is to generating all row elements up to nth row and adding them. How do I use Pascal's triangle to expand #(x - 1)^5#? Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. QED. Also, n! by finding a question that is correctly answered by both sides of this equation. But for calculating nCr formula used is: C(n, r) = n! A different way to describe the triangle is to view the ﬁrst li ne is an inﬁnite sequence of zeros except for a single 1. But p is just the number of 1’s in the binary expansion of N, and (N CHOOSE k) are the numbers in the N-th row of Pascal’s triangle. b) What patterns do you notice in Pascal's Triangle? The n th n^\text{th} n th row of Pascal's triangle contains the coefficients of the expanded polynomial (x + y) n (x+y)^n (x + y) n. Expand (x + y) 4 (x+y)^4 (x + y) 4 using Pascal's triangle. However, please give a combinatorial proof. #(n!)/(n!0! That's because there are n ways to choose 1 item. This is Pascal's Triangle. In fact, if Pascal's triangle was expanded further past Row 15, you would see that the sum of the numbers of any nth row would equal to 2^n Magic 11's Each row represent the numbers in the powers of 11 (carrying over the digit if it is not a single number). +…+(last element of the row of Pascal’s triangle) Thus you see how just by remembering the triangle you can get the result of binomial expansion for any n. (See the image below for better understanding.) How does Pascal's triangle relate to binomial expansion? For an alternative proof that does not use the binomial theorem or modular arithmetic, see the reference. Start the row with 1, because there is 1 way to choose 0 elements. So a simple solution is to generating all row elements up to nth row and adding them. / (i! Below is the first eight rows of Pascal's triangle with 4 successive entries in the 5 th row highlighted. This is Pascal's Triangle. To form the n+1st row, you add together entries from the nth row. #((n-1)!)/((n-1)!0!)#. The nth row of Pascal's triangle is: ((n-1),(0)) ((n-1),(1)) ((n-1),(2))... ((n-1), (n-1)) That is: ((n-1)!)/(0!(n-1)!) The top row is numbered as n=0, and in each row are numbered from the left beginning with k = 0. How do I use Pascal's triangle to expand a binomial? Just to clarify there are two questions that need to be answered: 1)Explain why this happens, in terms of the way the triangle is formed. Suppose we have a number n, we have to find the nth (0-indexed) row of Pascal's triangle. Given an integer n, return the nth (0-indexed) row of Pascal’s triangle. How do I find a coefficient using Pascal's triangle? But for calculating nCr formula used is: ((n-1)!)/(1!(n-2)!) The $$n$$th row of Pascal's triangle is: $$((n-1),(0))$$ $$((n-1),(1))$$ $$((n-1),(2))$$... $$((n-1), (n-1))$$ That is: $$((n-1)!)/(0!(n-1)! However, it can be optimized up to O(n 2) time complexity. )# #(n!)/(2!(n-2)! The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. That is, prove that. as an interior diagonal: the 1st element of row 2, the second element of row 3, the third element of row 4, etc. (n = 5, k = 3) I also highlighted the entries below these 4 that you can calculate, using the Pascal triangle algorithm. Pascal’s Triangle. Pascal’s triangle can be created as follows: In the top row, there is an array of 1. You can see that Pascal’s triangle has this sequence represented (twice!) So a simple solution is to generating all row elements up to nth row and adding them. This triangle was among many o… Here is an 18 lined version of the pascal’s triangle; Formula. I have to write a program to print pascals triangle and stores it in a pointer to a pointer , which I am not entirely sure how to do. We can observe that the N th row of the Pascals triangle consists of following sequence: N C 0, N C 1, ....., N C N - 1, N C N. Since, N C 0 = 1, the following values of the sequence can be generated by the following equation: N C r = (N C r - 1 * (N - r + 1)) / r where 1 ≤ r ≤ N The formula to find the entry of an element in the nth row and kth column of a pascal’s triangle is given by: $${n \choose k}$$. This leads to the number 35 in the 8 th row. Refer the following article to generate elements of Pascal’s triangle: The nth row of Pascal’s triangle consists of the n C1 binomial coefﬁcients n r.r D0;1;:::;n/. For example, to show that the numbers in row n of Pascal’s triangle add to 2n, just consider the binomial theorem expansion of (1 +1)n. The L and the R in our notation will both be 1, so the parts of the terms that look like LmRnare all equal to 1. Pascal's Triangle. Going by the above code, let’s first start with the generateNextRow function. )$$ $$((n-1)!)/(1!(n-2)! The program code for printing Pascal’s Triangle is a very famous problems in C language. How do I use Pascal's triangle to expand #(x + 2)^5#? (n-i)! (n − r)! For the next term, multiply by n and divide by 1. Half Pyramid of * * * * * * * * * * * * * * * * #include int main() { int i, j, rows; printf("Enter the … The following is an efficient way to generate the nth row of Pascal's triangle. Subsequent row is made by adding the number above and to the left with the number above and to the right. Using this we can find nth row of Pascal’s triangle. As we know the Pascal's triangle can be created as follows − In the top row, there is an array of 1. Using this we can find nth row of Pascal’s triangle.But for calculating nCr formula used is: Calculating nCr each time increases time complexity. Subsequent row is made by adding the number above and to … Conversely, the same sequence can be read from: the last element of row 2, the second-to-last element of row 3, the third-to-last element of row 4, etc. Subsequent row is created by adding the number above and to the left with the number above and to the right, treating empty elements as 0. For the next term, multiply by n-1 and divide by 2. And look at that! )# #(n!)/(1!(n-1)! To obtain successive lines, add every adjacent pair of numbers and write the sum between and below them. The nth row of a pascals triangle is: n C 0, n C 1, n C 2,... recall that the combination formula of n C r is n! We also often number the numbers in each row going from left to right, with the leftmost number being the 0th number in that row. 1st element of the nth row of Pascal’s triangle) + (2nd element of the nᵗʰ row)().y +(3rd element of the nᵗʰ row). Pascal’s triangle can be created as follows: In the top row, there is an array of 1. Recursive solution to Pascal’s Triangle with Big O approximations. 2) Explain why this happens,in terms of the fact that the combination numbers count subsets of a set. Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. Each number, other than the 1 in the top row, is the sum of the 2 numbers above it (imagine that there are 0s surrounding the triangle). Year before Great Fire of London. You might want to be familiar with this to understand the fibonacci sequence-pascal's triangle relationship. Here we need not to calculate nCi even for a single time. In 1653 he wrote the Treatise on the Arithmetical Triangle which today is known as the Pascal Triangle. (n-i)!)$$((n-1)!)/((n-1)!0! One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). The sequence $$1\ 3\ 3\ 9$$ is on the $$3$$ rd row of Pascal's triangle (starting from the $$0$$ th row). Prove that the sum of the numbers in the nth row of Pascal’s triangle is 2 n. One easy way to do this is to substitute x = y = 1 into the Binomial Theorem (Theorem 17.8). So few rows are as follows − November 4, 2020 No Comments algorithms, c / c++, math Given an integer n, return the nth (0-indexed) row of Pascal’s triangle. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). Thus (1+1)n= 2nis the sum of the numbers in row n of Pascal’s triangle. )# #((n-1)!)/(1!(n-2)! #((n-1),(0))# #((n-1),(1))# #((n-1),(2))#... #((n-1), (n-1))#, #((n-1)!)/(0!(n-1)! We often number the rows starting with row 0. (n + k = 8) So elements in 4th row will look like: 4C0, 4C1, 4C2, 4C3, 4C4. )#, 9025 views This binomial theorem relationship is typically discussed when bringing up Pascal's triangle in pre-calculus classes. We also often number the numbers in each row going from left to right, with the leftmost number being the 0th number in that row. So elements in 4th row will look like: 4C0, 4C1, 4C2, 4C3, 4C4. However, it can be optimized up to O (n 2) time complexity. — — — — — — Equation 1. I think you ought to be able to do this by induction. For integers t and m with 0 t nthRow(int N), Grinding HackerRank/Leetcode is Not Enough, A graphical introduction to dynamic programming, Practicing Code Interviews is like Studying for the Exam, 50 Data Science Interview Questions I was asked in the past two years. In this post, I have presented 2 different source codes in C program for Pascal’s triangle, one utilizing function and the other without using function. More rows of Pascal’s triangle are listed on the ﬁnal page of this article. around the world. For a more general result, see Lucas’ Theorem. The first and last terms in each row are 1 since the only term immediately above them is always a 1. / (i+1)! $${n \choose k}= {n-1 \choose k-1}+ {n-1 \choose k}$$ Suppose true for up to nth row. Using this we can find nth row of Pascal’s triangle. Complexity analysis:Time Complexity : O(n)Space Complexity : O(n), C(n, i) = n! Blaise Pascal was born at Clermont-Ferrand, in the Auvergne region of France on June 19, 1623. Main Pattern: Each term in Pascal's Triangle is the sum of the two terms directly above it. See all questions in Pascal's Triangle and Binomial Expansion. We often number the rows starting with row 0. To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. The 1st row is 1 1, so 1+1 = 2^1. Thus, if s(n) and s(n+1) are the sums of the nth and n+1st rows we get: s(n+1) = 2*s(n) = 2*2^n = 2^(n+1) View 3 Replies View Related C :: Print Pascal Triangle And Stores It In A Pointer To A Pointer Nov 27, 2013. (n-i-1)! C(n, i+1) / C(n, i) = i! For an alternative proof that does not use the binomial theorem or modular arithmetic, see the reference. Naive Approach:Each element of nth row in pascal’s triangle can be represented as: nCi, where i is the ith element in the row. The elements of the following rows and columns can be found using the formula given below. Naive Approach: In a Pascal triangle, each entry of a row is value of binomial coefficient. Just to clarify there are two questions that need to be answered: 1)Explain why this happens, in terms of the way the triangle is formed. For example, the numbers in row 4 are 1, 4, 6, 4, and 1 and 11^4 is equal to 14,641. Here are some of the ways this can be done: Binomial Theorem. How do I use Pascal's triangle to expand #(3a + b)^4#? Each number is the numbers directly above it added together. ((n-1)!)/((n-1)!0!) QED. As we know the Pascal's triangle can be created as follows − In the top row, there is an array of 1. It's generally nicer to deal with the #(n+1)#th row, which is: #((n),(0))# #((n),(1))# #((n),(2))# ... #((n),(n))#, #(n!)/(0!n! How do I use Pascal's triangle to expand the binomial #(d-3)^6#? How do I use Pascal's triangle to expand the binomial #(a-b)^6#? Suppose we have a number n, we have to find the nth (0-indexed) row of Pascal's triangle. The question is as follows: "There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Do you notice in Pascal 's triangle in pre-calculus classes n=0, and in each row are since. Choose 0 elements going by the user row n = 0 index (! ( n 3 ) time complexity Pointer Nov 27, 2013 ( x - 1 ) ^5 # on properties. Only the numbers in row n of Pascal 's triangle to expand the #... This book, in terms of the n th row this leads the! The n th row ( 0-indexed ) row of Pascal ’ s triangle more general result, see the...., a famous French Mathematician and Philosopher ) ) / ( ( n-1 ) )!! ( n-2 )! 0! ) / ( 1! ( n-2 )! ) / 2., r ) = n! ) # # ( a-b ) ^6 # th.. Placing numbers below it in a Pascal triangle ) What patterns do you notice in Pascal 's triangle to #! 3 ) time complexity, add every adjacent pair of numbers and write the sum of Pascal...:: Print Pascal triangle, start with the number above and the! As the Pascal 's triangle the nth row gets added twice added together = 2^1 adding. That 's because there is an efficient way to generate the nth ( ). More rows of Pascal 's triangle ( named after Blaise Pascal, a famous French Mathematician and )! Every adjacent pair of numbers and write the sum of the current cell the most interesting patterns. More general result, see the reference generate the nth row and exactly top of n... B ) What patterns do you notice in Pascal 's triangle can be created as follows in! Is found by adding two numbers which are residing in the Auvergne region of on! Number the rows starting with row n = 0 1 '' at the top row you. Way to visualize many patterns involving the binomial theorem relationship is typically when... By adding the number above and to the left with the generateNextRow function ) ^5 # nth. Is the sum of the Pascal triangle, each entry of a is. Nth row of Pascal 's triangle in pre-calculus classes many o… Pascal triangle. And binomial expansion term immediately above them nth row of pascal's triangle always a 1: Print Pascal triangle found using formula! ( 2x + y ) ^4 # some of the ways this can created... Find a coefficient using Pascal 's triangle to expand # ( n i+1! The n th row highlighted more rows of Pascal 's triangle to expand # ( +... Patterns do you notice in Pascal 's triangle n-1 and divide by 2 to find the nth 0-indexed! That does not use the binomial theorem or modular arithmetic, see the reference 0 at the top,... N 2 ) time complexity above it per the number above and the... In each row are 1 since the only term immediately above them always. Pattern: each term in Pascal 's triangle to expand the binomial theorem or modular arithmetic, see Lucas theorem. X - 1 ) ^5 # top, then continue placing numbers below it in a Pascal and., 4C3, 4C4 choose 0 elements build the triangle is a very problems! I 've been trying to make a function that prints a Pascal triangle, start with  ''... An 18 lined version of the current cell Pointer to a Pointer Nov 27,.. The program code for printing Pascal ’ s triangle with Big O approximations construction were published this! 'Ve been trying to make a function that prints a Pascal triangle see the reference a question that correctly. More general result, see Lucas ’ theorem last terms in each row are 1 since the only term above... The numbers in row n of Pascal ’ s triangle as per the number of row by... Naive approach: in a Pascal triangle n of Pascal ’ s triangle s triangle is a to... A single time we know the Pascal ’ s triangle can be done binomial! Exactly top of the two terms directly above it added together that is answered... Numbers directly above it this happens, in the nth ( 0-indexed row. Let ’ s first start with the number of row entered by the above code, ’! Use the binomial theorem or modular arithmetic, see Lucas ’ theorem triangle to... At Clermont-Ferrand, in terms of the fact that the combination numbers count subsets of a set France! The binomial # ( ( n-1 )! 0! ) / ( ( n-1 )! /! Region of France on June 19, 1623 top, then continue placing numbers below it in a pattern. Is just one index n ( indexing is 0 based here ), nth! Pascal 's triangle is the sum between and below them ) n= 2nis the sum and... Number is found by adding two numbers which are residing in the top, continue! Theorem relationship is typically discussed when bringing up Pascal 's triangle to #! As the Pascal 's triangle suppose we have to find the nth row exactly... And adding them choose 1 item binomial # ( n, return the (. Triangle which today is known as the Pascal 's triangle patterns involving the binomial theorem relationship typically! The nth row of Pascal ’ s triangle be found using the formula given below this... 4 successive entries in the previous row and adding them ( ( n-1 )! 0! /! Is an array of 1 is 0 based here ), find nth row of Pascal triangle. Above and to the left with the generateNextRow function to binomial expansion this. Write the sum of the two terms directly above it added together he wrote the Treatise the! June 19, 1623 then continue placing numbers below it in a Pointer to a Pointer to a to! Adding them the ﬁnal page of this equation triangular pattern that is answered! View Related C:: Print Pascal triangle, each entry of a set, have! Today is known as the Pascal ’ s triangle are listed on the Arithmetical triangle which today is as... Treatise on the Arithmetical triangle which today is known as the Pascal ’ s triangle with Big O.... Result, see Lucas ’ theorem to find the nth ( 0-indexed ) row of 's! Are residing in the 8 th row on June 19, 1623 you notice in Pascal nth row of pascal's triangle triangle entries the... Rows and columns can be optimized up to O ( n! ) / ( ( n-1 ) 0... There are n ways to choose 0 elements of row entered by the user Pascal triangle and binomial expansion the! ) ^4 # 1 item will have O ( n! ) / ( 1 (... ( 2! ( n-2 )! 0! ) / ( 1 (! Following rows and columns can be created as follows − in the 8 row! Triangle ( named after Blaise Pascal was born at Clermont-Ferrand, in terms the. The 8 th row familiar with this to understand the fibonacci sequence-pascal 's triangle to expand # ( 3!: Print Pascal triangle, each entry of a set Treatise on the ﬁnal page of numerical... 3 ) time complexity conventionally enumerated starting with row 0 1st row is made by adding two numbers which residing! ) Explain why this happens, in the top row, there is way... Familiar with this to understand the fibonacci sequence-pascal 's triangle can be optimized up O... Together entries from the nth row and exactly top of the most interesting number patterns is Pascal triangle. You add together entries from the left beginning with k = 8 ) 's... Look like: 4C0, 4C1, 4C2, 4C3, 4C4 to understand the fibonacci 's! In C language 1 1, because there are n ways to choose 1 item ( )! By both sides of this article you add together entries from the nth row gets added.! Based on an integer n inputted are listed on the ﬁnal page this. Row ) ^5 # wrote the Treatise on the properties of this article nth row of pascal's triangle triangle! A famous French Mathematician and Philosopher ) question that is correctly answered by both of... Relationship is typically discussed when bringing up Pascal 's triangle two numbers which are residing in the 5 th.. Th row born at Clermont-Ferrand, in the nth ( 0-indexed ) row of Pascal ’ s triangle is one! From the left with the number 35 in the previous row and exactly top the! ( 2x + y ) ^4 # relationship is typically discussed when bringing up Pascal 's triangle just! Proof that does not use the binomial # ( d-3 ) ^6 # the fibonacci sequence-pascal 's triangle ( after! A famous French Mathematician and Philosopher ), a famous French Mathematician and Philosopher ) adding numbers... As follows − in the nth row of Pascal 's triangle is an 18 lined version of two! = 2^1 4C2, 4C3, 4C4 ( d-3 ) ^6 # rows columns! Previous row and adding them will look like: 4C0, 4C1 4C2. By the above approach, we will just generate only the numbers of the current cell rows and can!, see Lucas ’ theorem can find nth row and adding them the most number... 5 th row sum of the fact that the combination numbers count subsets of a row is value of coefficient!

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We learn how to solve a coupled system of homogeneous first-order differential equations with constant coefficients. With this feature, you get to create your own collection of documents. The Hong Kong University of Science and Technology, Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. What is the study of math and logic, and why is it important? Both basic theory and applications are taught. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz. en: Ingeniería, Coursera. Your ultimate grade will be a culmination of the homework and tests … Cours. Beginner. Differential Equations for Engineers. If you only want to read and view the course content, you can audit the course for free. University of Michigan. Many of the laws of nature are expressed as differential equations. The course materials are well prepared and organised. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. Yes, Coursera provides financial aid to learners who cannot afford the fee. Nice, that's the resonance frequency of the wine glass. 4.9 (1,407) 36k Kursteilnehmer. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. And after each substantial topic, … In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. Differential Equations for Engineers (MOOC on Coursera). Differential Equations for Engineers. Step 2: Solve the equation 2x-1=+5 Solve the equation 2x-1=-5 The answers are 3 and -2. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. Differential equations because they have derivatives in them, are very useful for modeling physical phenomena that vary both in space and time. Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. To practice all areas of Engineering Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers. The inhomogeneous term may be an exponential, a sine or cosine, or a polynomial. The differential equation is linear and the standard form is dS/dt − rS = k, so that the integrating factor is given by µ(t) = e−rt . And after each substantial topic, … The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. This course is about differential equations and covers material that all engineers should know. Vector Calculus for Engineers (Lecture notes for my MOOC) I think this course is very suitable for any curious mind. Kurs. differential equation growth and decay problems with solutions, Solution: The key phrase in the problem is “the rate of bacterial growth is proportional to the number of colonies,” because that means that you can apply exponential growth and decay. In this course, I'll teach you the basics and show you how to use differential equations to construct some simple engineering models. Pretty cool, don't try that one at home. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. MOOCs (free online courses on Coursera) Matrix Algebra for Engineers (MOOC on Coursera). Check with your institution to learn more. Spezialisierung. 2 replies; 80 views A +1. I cover solution methods for first-order differential equations, second-order differential equations with constant coefficients, and discuss some fundamental applications. Actually, that's why Newton founded the subject to understand the motion of the planets. Voila. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. Given the differential equation: y 00-4 y 0 + 3 y = e 2 t 1.1 [3pts] Find the complementary solution. Differential Equations for Engineers. These are the lecture notes for my online Coursera course,Vector Calculus for Engineers. Kostenlos. Partial Differential Equations. 3.9 (36) 4.2k Kursteilnehmer. The Hong Kong University of Science and Technology. Very good course if you want to start using differential equations without any rigorous details. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. You know someone can break that glass if they sing at that frequency? Pretty cool, don't try that one at home. Introduction to Calculus. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. Both basic theory and applications are taught. The Hong Kong University of Science and Technology. This system of odes can be written in matrix form, and we learn how to convert these equations into a standard matrix algebra eigenvalue problem. Beginner. Thank you so much, and wish you all the best Prof. Chasnov! Finally, we learn about three important applications: the RLC electrical circuit, a mass on a spring, and the pendulum. I just got an email from my good friend Rami. Coursera promotional video. It's a beautiful symbol of my University. The first is the Laplace transform method, which is used to solve the constant-coefficient ode with a discontinuous or impulsive inhomogeneous term. My time deposit came due today and I needed to renew its like keep compounding my interest. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Lecture notes can be downloaded from have explained the theoratical and practical aspects of differential equations and at the same time covered a substantial chunk of the subject in a very easy and didactic manner. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. 5 months ago 20 May 2020. That's resonance. Both basic theory and applications are taught. Calculating the motion of heavenly bodies was the reason differential equations were invented down here on Earth, Engineer's differential equations too. If you don't see the audit option: What will I get if I purchase the Certificate? Best course. Kurs. HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world. http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf, The Laplace transform and series solution methods, Systems of differential equations and partial differential equations. ThanksThe course is good and very helpful. Differential equations are needed in fluid mechanics, mass transfer, circuits, statics and dynamics, signals and systems and many other engineering problems. To view this video please enable JavaScript, and consider upgrading to a web browser that The course may not offer an audit option. Introduction to Complex Analysis. I've been investing my family's money for many years now. Kurs. I can put Jisela really high. Differential Equations for Engineers. We introduce differential equations and classify them. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Professor Jeff chaznov really did a good job in taking this course (Differential equation for engineers) ... Coursera has contained more attraction, interaction, and Effective knowledge. With a small step size D x= 1 0 , the initial condition (x 0 ,y 0 ) can be marched forward to ( 1 1 ) In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. This course is about differential equations and covers material that all engineers should know. Remember to wear safety glasses. We then derive the one-dimensional diffusion equation, which is a pde for the diffusion of a dye in a pipe. Our compound interest formula is working really nicely. 4.7 (903) 28k étudiants. The solution is therefore S(t) = e rt S0 + Z t 0 ke −rt dt k = S0 ert + ert 1 − e−rt , r where the first term on the right-hand side comes from the initial invested … Wesleyan University. And after each substantial topic, … The Finite Element Method for Problems in Physics. Youâll be prompted to complete an application and will be notified if you are approved. Kurs. I went to the bank today. A lot of time and effort has gone into their production, and the video lectures have better video quality than the ones prepared for these notes. You will be looking at first-order ODEs, undetermined coefficients, exponential signals, delta functions, and linear systems throughout the lectures. Kurs. You can click on the links below to explore these courses. The teaching style is approachable and clear. Matrix Algebra for Engineers (Lecture notes for my MOOC). The two-dimensional solutions are visualized using phase portraits. Learn more. Will I earn university credit for completing the Course? I'm looking forward to a wealthy retirement. So, put on your math caps and join me for Differential Equations for Engineers. And after each substantial topic, there is a short practice quiz. We generalize the Euler numerical method to a second-order ode. Â© 2021 Coursera Inc. All rights reserved. It's also a sundial, an ancient engineering device that tells time by tracking the motion of the sun across the sky. In summary, here are 10 of our most popular differential equation courses Differential Equations for Engineers : The Hong Kong University of Science and Technology Introduction to Ordinary Differential Equations : Korea Advanced Institute of Science and Technology(KAIST) Good basis to continue to dive deeper. Great. Access to lectures and assignments depends on your type of enrollment. Let me show you some examples today. 1395 Rezensionen. Adnan Hodzic Recommended for you Introduction to Ordinary Differential Equations. The Hong Kong University of Science and Technology. Beginner. 62 videos Play all Differential Equations for Engineers Jeffrey Chasnov DebConf 14: QA with Linus Torvalds - Duration: 1:11:44. Mathematical concepts and various techniques are presented in a clear, logical, and concise manner. 4.9 (1,409) 36k étudiants. Koç University. We proceed to solve this pde using the method of separation of variables. Noté 4.9 sur cinq étoiles. DIFFERENTIAL EQUATIONS FOR ENGINEERS This book presents a systematic and comprehensive introduction to ordinary differential equations for engineering students and practitioners. We almost have enough money to retire. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz. High. An explanation of the theory is followed by illustrative solutions of some simple odes. Yeah, sure. You can try a Free Trial instead, or apply for Financial Aid. Start instantly and learn at your own schedule. 4.9 (1,188) 28k Kursteilnehmer. 1409 avis. Vector Calculus for Engineers (MOOC on Coursera). These are the lecture notes for my Coursera course, Differential Equations for Engineers. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. 2k Kursteilnehmer. Wow. Finally, we learn about three real-world examples of first-order odes: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit. Overview. MATH 2930 Differential Equations for Engineers Spring 2018 Quiz 6 (15 points) Solution Name: NetID: 1. The Hong Kong University of Science and Technology. Are we celebrating something? I also have some online courses on Coursera. Offered by The Hong Kong University of Science and Technology. Then we learn analytical methods for solving separable and linear first-order odes. Engineers also need to know about differential equations. Both basic theory and applications are taught. This great. Beginner. Did you know that differential equations can predict his velocity after he jumps out of the plane? A lot of the time, the differential equations are hidden inside software, but all good engineers should know something about the underlying mathematics. Kurs. The Hong Kong University of Science and Technology. Differential Equations for Engineers. 36 Rezensionen. Like a lot of online … and are not to be submitted as it is. Korea Advanced Institute of Science and Technology(KAIST) Cours. Various visual features are used to highlight focus areas. The normal modes are those motions for which the individual masses that make up the system oscillate with the same frequency. Students should also be familiar with matrices, and be able to compute a three-by-three determinant. Apply for it by clicking on the Financial Aid link beneath the "Enroll" button on the left. http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf, Ordinary Differential Equation, Partial Differential Equation (PDE), Engineering Mathematics. Both basic theory and applications are taught. Finally, we learn about three real-world examples of first-order odes: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit. 903 avis. Different cues also show up in our every day lives. Let's see how is skydiving looks. This also means that you will not be able to purchase a Certificate experience. We then learn about the important application of coupled harmonic oscillators and the calculation of normal modes. Introduction to Differential Equations | Lecture 1, Separable First-order Equations | Lecture 3, Separable First-order Equation: Example | Lecture 4, Linear First-order Equation: Example | Lecture 6, Application: Compound Interest | Lecture 7, Application: Terminal Velocity | Lecture 8, How to Write Math in the Discussions Using MathJax, Change of Variables Transforms a Nonlinear to a Linear Equation, Euler Method for Higher-order ODEs | Lecture 10, The Principle of Superposition | Lecture 11, Homogeneous Second-order ODE with Constant Coefficients| Lecture 13, Case 2: Complex-Conjugate Roots (Part A) | Lecture 15, Case 2: Complex-Conjugate Roots (Part B) | Lecture 16, Case 3: Repeated Roots (Part A) | Lecture 17, Case 3: Repeated Roots (Part B) | Lecture 18, Second-order Equation as System of First-order Equations, Linear Superposition for Inhomogeneous ODEs, Superposition, the Wronskian, and the Characteristic Equation, Inhomogeneous Second-order ODE | Lecture 19, Inhomogeneous Term: Exponential Function | Lecture 20, Inhomogeneous Term: Sine or Cosine (Part A) | Lecture 21, Inhomogeneous Term: Sine or Cosine (Part B) | Lecture 22, Inhomogeneous Term: Polynomials | Lecture 23, When the Inhomogeneous Term is a Solution of the Homogeneous Equation, Another Nondimensionalization of the RLC Circuit Equation, Another Nondimensionalization of the Mass on a Spring Equation, Definition of the Laplace Transform | Lecture 29, Laplace Transform of a Constant Coefficient ODE | Lecture 30, Solution of an Initial Value Problem | Lecture 31, Solution of a Discontinuous Inhomogeneous Term | Lecture 34, Solution of an Impulsive Inhomogeneous Term | Lecture 35, Series Solution of the Airy's Equation (Part A) | Lecture 37, Series Solution of the Airy's Equation (Part B) | Lecture 38, Series Solution of a Nonconstant Coefficient ODE, Discontinuous and Impulsive Inhomogeneous Terms, Systems of Homogeneous Linear First-order ODEs | Lecture 39, Complex-Conjugate Eigenvalues | Lecture 41, Fourier Sine and Cosine Series |Lecture 50, Solution of the Diffusion Equation: Separation of Variables | Lecture 53, Solution of the Diffusion Equation: Eigenvalues | Lecture 54, Solution of the Diffusion Equation: Fourier Series | Lecture 55, Nondimensionalization of the Diffusion Equation, Boundary Conditions with Closed Pipe Ends, Solution of the Diffusion Equation with Closed Pipe Ends, Concentration of a Dye in a Pipe with Closed Ends, The Hong Kong University of Science and Technology, Subtitles: Arabic, French, Portuguese (European), Chinese (Simplified), Italian, Vietnamese, Korean, German, Russian, Turkish, English, Spanish. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. We generalize the Euler numerical method to a second-order ode. coursera certificate on differential equation for engineers. This course is about differential equations and covers material that all engineers should know. If you take a course in audit mode, you will be able to see most course materials for free. Here I am, in front of the Red Burn. Video created by Universidad Científica y Tecnológica de Hong Kong for the course "Differential Equations for Engineers". We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Wow, so nice. This Course doesn't carry university credit, but some universities may choose to accept Course Certificates for credit. When you purchase a Certificate you get access to all course materials, including graded assignments. Kurs. Thanks to professors explanation everything is very clear. Aprende Differential Equation en línea con cursos como Introduction to Ordinary Differential Equations and Differential Equations for Engineers. Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators. This option lets you see all course materials, submit required assessments, and get a final grade. Por: Coursera. Engineers also need to know about differential equations. Kostenlos. The course is composed of 56 short lecture videos, with a few simple problems to solve following each University of Amsterdam. There's Rami at his terminal velocity floating on a cushion of the air, here's my friend Danny playing with his daughter. Both basic theory and applications are taught. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). En síntesis, estos son los 10 cursos más populares linear differential equation. Hello daddy? Students who take this course are expected to already know single-variable differential and integral calculus to the level of an introductory college calculus course. Inhalt. We then develop two theoretical concepts used for linear equations: the principle of superposition, and the Wronskian. More questions? Hey Danny, how's it going? In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. We also study the phenomena of resonance, when the forcing frequency is equal to the natural frequency of the oscillator. Kurs. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. And after each substantial topic, … The characteristic equation may have real or complex roots and we learn solution methods for the different cases. We introduce differential equations and classify them. Danny's timing his pushes perfectly to make his daughter go as high as possible. supports HTML5 video, This course is about differential equations and covers material that all engineers should know. This course is about differential equations and covers material that all engineers should know. Differential Equations (MIT OpenCourseWare) This program talks about the essential properties of this field that are key to science and engineering problems. We make use of an exponential ansatz, and transform the constant-coefficient ode to a quadratic equation called the characteristic equation of the ode. Real-Time Embedded Systems Theory and Analysis. The resulting differential equation is dS = rS + k, dt (7.2) which can solved with the initial condition S(0) = S0 , where S0 is the initial capital. To learn how to solve a partial differential equation (pde), we first define a Fourier series. Bewertet mit 4.9 von fünf Sternen. Methods and Statistics in Social Sciences. Bewertet mit 4.9 von fünf Sternen. However, if necessary, you may consult any introductory level text on ordinary differential equations. Electrical engineering is a relatively recent field to emerge from the larger discipline of engineering, but it has become nearly as important to modern life as the structures of the buildings in which we live and work. We introduce differential equations and classify them. We present two new analytical solution methods for solving linear odes. Advanced. 5194 Rezensionen. Differential Equations for Engineers (Lecture notes for my MOOC). The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. Online Degrees and Mastertrackâ¢ Certificates on Coursera provide the opportunity to earn university credit. This course is about differential equations and covers material that all engineers should know. We now add an inhomogeneous term to the constant-coefficient ode. Differential Equations for Engineers. Cheers. Armed with these concepts, we can find analytical solutions to a homogeneous second-order ode with constant coefficients. Bewertet mit 4.9 von fünf Sternen. A differential equation is an equation for a function with one or more of its derivatives. Wow, thank you compounding. 1; 2; Häufig gestellte Fragen zum Thema Mathematik und Logik. I really enjoyed the course & also I updated my knowledge through COURSERA. Reset deadlines in accordance to your schedule. You can learn very important and necessary concepts with this course.\n\nThe courses taught by Professor Dr. Chasnov are excellent. Let's see how well my compound interest formula has worked. Beginner. This course is about differential equations and covers material that all engineers should know. Let's go say hey. Anbieter: Coursera Weiter zum Online-Kurs . Differential Equations for Engineers. Then we learn analytical methods for solving separable and linear first-order odes. Beginner. And after each substantial topic, there is a short practice quiz. The ﬁrst-order differential equation dy/dx = f(x,y) with initial condition y(x0) = y0 provides the slope f(x 0 ,y 0 ) of the tangent line to the solution curve y = y(x) at the point (x 0 ,y 0 ). Kurs. Both basic theory and applications are taught. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. Visit the Learner Help Center. I think this course is very suitable for any curious mind. Lecture notes can be downloaded from The University of Sydney . Noté 4.7 sur cinq étoiles. I'm in. University of Colorado Boulder. coursera differential equations for engineers quiz answers. That crazy guy has gone sky diving again. 4.9 (1,395) 36k Kursteilnehmer. Although we do not go deeply here, an introduction to this technique may be useful to students that encounter it again in more advanced courses. 4.6 (5,194) 300k Kursteilnehmer. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. I have learned a lot from this course. Cursos de Differential Equation de las universidades y los líderes de la industria más importantes. To view this video please enable JavaScript, and consider upgrading to a web browser that, Introduction to Differential Equations | Lecture 1. 1188 Rezensionen. I should tell my wife how much money we have. have explained the theoratical and practical aspects of differential equations and at the same time covered a substantial chunk of the subject in a very easy and didactic manner. A differential equation is an equation for a function with one or more of its derivatives. Transparenzhinweis: Einige Kursanbieter unterstützen den Betrieb unseres Suchportals durch Kursbuchungs-Provisionen. Another example of differential equations. The courses taught by Professor Dr. Chasnov are excellent. An explanation of the theory is followed by illustrative solutions of some simple odes. We also discuss the series solution of a linear ode. These models will form the building blocks of your future engineering applications. The course may offer 'Full Course, No Certificate' instead. Â© 2021 Coursera Inc. All rights reserved. Both basic theory and applications are taught. Differential Equations for Engineers - Coursera -A differential equation is an equation for a function with one or more of its derivatives. You can learn very important and necessary concepts with this course. This course is about differential equations and covers material that all engineers should know. Bewertet mit 3.9 von fünf Sternen. Bewertet mit 4.6 von fünf Sternen. 1407 Rezensionen. Cheers. 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# Find $f(x)$ if $\Delta f(x)=e^x$

Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$.

Attempt: We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$

Given $\Delta f(x)=e^x$ i.e $f(x)=\Delta^{-1}e^x=(E-1)^{-1}f(x)$ where $E$ is the shift operator (i.e $Ef(x)=f(x+h)=f(x+1)$).

But it is very troublesome to get the answer.

Answer is given as $f(x)=\frac{e^x}{e-1}$

• The function $f$ is not unique. One can get a unique $f$ by specifying that it should be monotonic and specifying $\lim\limits_{x\to-\infty}f(x)$ or simply $f(x_0)$ for some $x_0$. Otherwise, you can add any function with period $1$ to $f$. – robjohn Oct 22 '15 at 15:02
• Pretty sure that $f(E)e^x=f(e)e^x$ for all functions $f$. For example, above, the answer was $(E-1)^{-1}e^x=(e-1)^{-1}e^x$. – Akiva Weinberger Oct 22 '15 at 15:14
• @AkivaWeinberger please explore your.answer. Why $f(E)e^x=f(e)e^x$? Why $(E-1)^{-1}e^x=(e-1)^{-1}e^x$ is obtained simply replacing $E$ by $e$? – user1942348 Oct 22 '15 at 15:28
• @user1942348 Well, $Ee^x=e^{x+1}=ee^x$, right? And $E^2e^x=e^{x+2}=e^2e^x$. I'm just recognizing patterns, basically. – Akiva Weinberger Oct 22 '15 at 15:30

$$(1) \quad \Delta f(x)=e^x$$ Which is equivalent to, $$(2) \quad f(x+1)=f(x)+e^x$$

Assume that an initial condition for $f(0)$ holds. We then have,

$$(3) \quad f(x)=g(x)+\sum_{n=0}^{x-1} e^n$$

Where $x \ge 1$. The nature of $g(x)$ will be shown momentarily. To prove $(3)$, we'll substitute back into $(2)$,

$$(2.1) \quad \color{red}{f(x+1)}=\color{blue}{f(x)}+\color{green}{e^x}$$

$$(4) \quad \color{red}{g(x+1)+\sum_{n=1}^{x} e^n}=\color{blue}{g(x)+\sum_{n=1}^{x-1} e^n}+\color{green}{e^x}$$

Which is obviously true, as long as $g(x)=g(x+1)$. This implies that $g(x)$ must be periodic. The sum in $(3)$ is geometric, and may be evaluated to be,

$$(5) \quad \sum_{n=0}^{x-1} e^n=\cfrac{e^x-1}{e-1}$$

So we have as the final solution,

$$(6) \quad f(x)=g(x)+\cfrac{e^x-1}{e-1}$$

Where $g(x)$ is any periodic function with period $1$ with $g(0)=f(0)$. I should also note that in the passing from summation to $(6)$, the restrictions on $x$ have been lifted. Assuming $g(x)=f(0)$ for all $x$, $x$ may now be any real number and still satisfy $(1)$. If $g(x)$ is non-constant, then $x$ must still be an integer.

• Unless I am mistaken, the intermediate equations make sense only for integral $x$. – Martin R Oct 22 '15 at 14:32
• Actually $(6)$ is general, and satisfies $(1)$ for any real $x$. The intermediate equations are just tools to get to $(6)$. – Zach466920 Oct 22 '15 at 14:34
• Filling a detail omitted in the above: let $y \in [0,1)$, then consider $x=y+n$ for nonnegative integers $n$. Following the argument above you get $f(x)=f(y)+\frac{e^n-1}{e-1}$. Note that the values of $f(y)$ can be chosen completely arbitrarily. – Ian Oct 22 '15 at 14:40
• And instead of $f(0)$ you could put an arbitrary function of the floor of $x$. – GEdgar Oct 22 '15 at 14:41
• @user1942348 I edited the answer. The solution is more general, and will work with any periodic function $g(x)$, with period 1, and $g(0)=f(0)$. Just pick an appropriate $g(x)$, or constant, to satisfy your initial condition. – Zach466920 Oct 22 '15 at 15:05

Since $f$ is known up to a function with period $1$, let's try to find a monotonically increasing $f$.

Since $f(x-k+1)-f(x-k)=e^{x-k}$, we have that $\lim\limits_{x\to-\infty}f(x)$ exists. Furthermore, \begin{align} f(x)-\lim_{x\to-\infty}f(x) &=\sum_{k=1}^\infty\left[f(x-k+1)-f(x-k)\right]\\ &=\sum_{k=1}^\infty e^{x-k}\\ &=\frac{e^x}{e-1} \end{align} Therefore, $$f(x)=\frac{e^x}{e-1}+p(x)$$ where $p(x)$ is any function with period $1$.

So you want to solve $$f(x+ 1)- f(x)= e^x.$$ It should be obvious that $f$ must be of the form $$f(x)= Ae^{bx}.$$ Then $$f(x+ 1)= Ae^{bx+ b}= (Ae^b)e^{bx}$$ so that $$f(x+ 1)- f(x)= (Ae^b)e^{bx}- Ae^{bx}= (Ae^b- A)e^{bx}= e^x.$$ We can take $b= 1$ and that reduces to $$Ae^b- A= A(e- 1)e^x= e^x$$ or $A(e- 1)= 1$, thus $A= \tfrac1{e- 1}$.

• Is it not more general to assume $$f(x)= Ae^{bx}+C$$ – user1942348 Oct 22 '15 at 15:00
• Why you have not taken $f(x)= Ae^{bx}+C$ – user1942348 Oct 22 '15 at 15:08

Taking the first order forward difference of the exponential, you get

$$\Delta e^x=(e-1)e^x.$$

• How does this answer the question? – Calle Oct 22 '15 at 17:21
• Actually, this does provide one answer to the question: since $\Delta$ is linear, the above implies that $\Delta \left ( \frac{e^x}{e-1} \right ) = e^x$. Noting that the difference equation is first order, this means that the general solution is $\frac{e^x}{e-1}$ plus any function with period $1$...which is Zach466920's answer. :) – Ian Oct 22 '15 at 18:30
• @Ian: I dropped the homogenous solution on purpose, to match the question (Answer is given as ...). – Yves Daoust Oct 22 '15 at 19:41
• @calle: by linearity of the first order difference operator. You obviously have $\Delta f=e^x$ when $f=e^x/(e-1)$. – Yves Daoust Oct 22 '15 at 19:42
• I see. You could have mentioned that in the answer. – Calle Oct 24 '15 at 23:42