# Induction Proof Statement Help: x^n-y^n = (x-y)*sum

• hicsuntdrac0ni
In summary: If you substitute your supposed formula for the first n terms into the sum, you get something that doesn't match what you get when you substitute it into the expanded expression on the right side.In summary, the formula for the sum of the first n terms of the expression x^n-y^n is (x^n-y^n)/(x-y), and when n is increased by 1, the resulting sum is (x-y)(x^n+x^(n-1)y+...+xy^(n-1)+y^n). This can be seen by expanding the expression on the right side.
hicsuntdrac0ni

## Homework Statement

https://sphotos-b.xx.fbcdn.net/hphotos-prn1/69668_10151632316928154_624610826_n.jpg x^n-y^n = (x-y)*(x^(n-1)+x^(n-2)*y+...+x*y^(n-2)+y^(n-1))

from Number Theory by George E. Andrews

## The Attempt at a Solution

(x^n-y^n)/(x-y) = the sum for the first n numbers and then i added (x*y^((n+1)-2)+y^((n+1)-1)) which should equal (x^(n+1)-y^(n+1))/(x-y) but i can't figure it out

Last edited by a moderator:
That's not right. If you let ##n \rightarrow n+1##, you get
\begin{align*}
x^{n+1}-y^{n+1} &= (x-y)(x^{(n+1)-1}+x^{(n+1)-2}y+\cdots+xy^{(n+1)-2}+y^{(n+1)-1}) \\
&= (x-y)(x^n+x^{n-1}y+\cdots+xy^{n-1}+y^{n})
\end{align*} You tacked on the last two terms, but all of the other terms in the sum don't match.

vela said:
That's not right. If you let ##n \rightarrow n+1##, you get
\begin{align*}
x^{n+1}-y^{n+1} &= (x-y)(x^{(n+1)-1}+x^{(n+1)-2}y+\cdots+xy^{(n+1)-2}+y^{(n+1)-1}) \\
&= (x-y)(x^n+x^{n-1}y+\cdots+xy^{n-1}+y^{n})
\end{align*} You tacked on the last two terms, but all of the other terms in the sum don't match.

The expressions after the "+...+" are supposed to be the nth term so the n+1 term should be those last two with (n+1) substituted for (n) . Since the first nth terms are x^(n-1)+x^(n-2)*y+...+x*y^(n-2)+y^(n-1) and that is equal to x^n-y^n/(x-y) then I can substitute the x^(n-1)+x^(n-2)*y+...+x*y^(n-2)+y^(n-1) for x^n-y^n/(x-y) and then add x^(n+1)-y^(n+1)/(x-y)

I just showed you the first n-1 terms in the sum aren't what you think they're equal to.

## 1. What is an induction proof statement?

An induction proof statement is a mathematical proof that uses the principle of mathematical induction to establish the truth of a statement for all natural numbers. It involves showing that a statement holds for a base case (usually n = 1) and then using the inductive hypothesis to show that it also holds for n+1.

## 2. What is the purpose of using induction in a proof?

The purpose of using induction in a proof is to establish the truth of a statement for all natural numbers, without having to explicitly prove it for each individual case. It allows for a more concise and efficient way of proving mathematical statements.

## 3. How do you prove x^n-y^n = (x-y)*sum using mathematical induction?

To prove this statement using mathematical induction, you would first show that it holds for the base case (n = 1). Then, assuming it holds for n, you would use the inductive hypothesis to show that it also holds for n+1. This would involve expanding the expression for (n+1) and simplifying it to match the original statement.

## 4. Can induction be used to prove any mathematical statement?

No, induction can only be used to prove statements that are true for all natural numbers. It cannot be used to prove statements that are only true for a finite number of cases, or for real numbers.

## 5. Are there any common mistakes to avoid when using induction in a proof?

One common mistake to avoid when using induction in a proof is assuming that the statement is true for all natural numbers without properly proving it for the base case and using the inductive hypothesis to show it holds for all subsequent cases. It is also important to make sure that the inductive step is correct and that the statement holds for n+1, not just for n.

Replies
8
Views
732
Replies
3
Views
965
Replies
7
Views
709
Replies
15
Views
2K
Replies
23
Views
1K
Replies
1
Views
960
Replies
7
Views
751
Replies
4
Views
905
Replies
1
Views
966
Replies
6
Views
1K