Proof by Induction

Statement

$$x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1})$$

Proof

Suppose n=1:

$$x - y = (x-y)*1 = x-y$$

Suppose the statement is true for all k < n:

$\quad (x^k - y^k)(x + y) = x^{k+1} + x^ky - xy^k - y^{k+1}$

$\quad x^{k+1} - y^{k+1} = (x^k - y^k)(x+y)-x^ky+xy^k$

$\quad x^{k+1} - y^{k+1} = (x-y)(x^{k-1} + x^{k-2}y \cdots + xy^{k-2} + y^{k-1})(x+y)-xy(x^{k-1}+y^{k-1})$

$\quad x^{k+1} - y^{k+1} = (x-y)(x^{k-1} + x^{k-2}y \cdots + xy^{k-2} + y^{k-1})(x+y)-xy(x^{k-1}-y^{k-1})$

$\quad x^{k+1} - y^{k+1} = (x-y)(x^{k-1} + x^{k-2}y \cdots + xy^{k-2} + y^{k-1})(x+y)-xy(x-y)(x^{k-2}+x^{k-3}y+\cdots+xy^{k-3}+y^{k-2})$

$\quad x^{k+1} - y^{k+1} = (x-y)(x^{k-1} + x^{k-2}y \cdots + xy^{k-2} + y^{k-1})(x+y)-(x-y)(x^{k-1}+x^{k-2}y^2+\cdots+x^2y^{k-2}+y^{k-1})$

$\quad x^{k+1} - y^{k+1} = (x-y)(x^k+x^{k-1}y+\cdots+x^2y^{k-2}+xy^{k-1}+x^{k-1}y+x^{k-2}y^2+\cdots+xy^{k-1}+y^k-x^{k-1}y-x^{k-2}y^2-\cdots-x^2y^{k-2}-xy^{k-1})$

$\quad x^{k+1} - y^{k+1} = (x-y)(x^k+x^{k-1}y+\cdots+x^2y^{k-2}+xy^{k-1}+y^k) \blacksquare$