Question

Prove by induction:
$x^n-y^n$ is divisible by $x+y$ when $n$ is even.

Answer 1

The statement to be proved is:
$P\left(n\right):x^n-y^n$ is divisble by $x+y$ where $n$ is even.

Step 1: Verify that the statement is true for the smallest value of $n$, here, $n=2$

$P\left(2\right):x^2-y^2$ is divisible by $x+y$

$P\left(2\right):(x+y)(x-y)$ is divisible by $x+y$, which is true.

Therefore $P\left(2\right)$ is true.

Step 2: Assume that the statement is true for $k$

Let us assume that $P\left(k\right):x^k-y^k$ is divisible by $x+y$ where $k$ is even.

Step 3: Verify that the statement is true for the next possible integer, here for $n=k+2$

$x^{k+2}-y^{k+2}=x^{k+2}-x^2{y}^k+x^2{y}^k-y^{k+2}$

$=x^2(x^k-y^k)+y^k(x^2-y^2)$

Since $(x^k-y^k)$ and $(x^2-y^2)$ are both divisible by $(x+y)$, the complete equality is divisible by $x+y$

Therefore,

$P(k+2):x^{k+2}-y^{k+2}$ is divisible by $x+y$ where $k+2$ is even.

Therefore by principle of mathematical induction, $P\left(n\right)$ is true.