A proof of a sum notation using induction - Need a hint.

Question

Heres a problem from the induction / sum notation part of the book im studying from. Can anyone hint me about how to prove it?

Show that for every $x,y$ : $$ (x^n-y^n) = (x-y)\sum_{i=1}^n x^{n-i} \cdot y^{i-1}$$

I tried to change the border of the sum to be i=n to 2n, and then use induction but it didn't seem to work.

Thanks.

Answer 1

If $n=1$, then all that the equality asserts is that $x-y=x-y$. Now, assuming that the equaliy holds for a certain natural $n$ (and for every $x$ and evey $y$), you can observe that\begin{align}x^{n+1}-y^{n+1}&=x^{n+1}-x^ny+x^ny-y^{n+1}\\&=x^n(x-y)+(x^n-y^n)y\end{align}and then you can use the induction hypothesis.

On the other hand, I would write the sum $\displaystyle\sum_{i=1}^nx^{n-i}y^{i-1}$ as $x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1}$.

Answer 2

A proof using induction always has two parts. Naturally, it starts with a statement about all integers, i.e. it has the form $\forall n\in\mathbb N: P(n)$

1. In the first part, you must prove that the statement is true for the integer $1$. In other words, you must prove $P(1)$.
2. In the second part, you can assume that the statement is true for $n$ (where $n$ is general), and from that, you must prove that the statement is true for $n+1$. In other words, you must prove $P(n)\implies P(n+1)$.

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I don't know exactly where you are stuck, so please, in the comment, answer these questions:

• Did you write down what the statement looks like for $n=1$? If so, what does it look like?
• Did you already prove the statement for $n=1$?
• Did you write down what the statement looks like for $n+1$? If so, what does it look like?

After you answer these, I can help you further.