mathematical induction

Let P(n): (x+y)^n - x^n - y^n is divisible by xy(x+y).

When n = 1: (x + y) - x - y = 0 which is divisible by xy(x + y)

Assume that P(k) is true, (x+y)^k - x^k - y^k is divisible by xy(x+y).

When n = k + 1

(x + y)^(k + 1) - x^(k + 1) - y^(k + 1)

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

= x[(x + y)^k - x^k - y^k] + y[(x + y)^k - x^k - y^k] + xy^k + yx^k

= x[Axy(x + y)] + y[Bxy(x + y)] + xy[x^(k - 1) + y^(k - 1)]

where A and B are the expressions of variables x and y

As x^(k - 1) + y^(k - 1) is divisible by (x + y), we just prove that

(x + y)^(k + 1) - x^(k + 1) - y^(k + 1) is divisible by (x + y).

So. P(k + 1) is true.

By M.I. for all positive integer n, (x+y)^n - x^n - y^n is divisible by xy(x + y)

@myisland8132
sorry!!! 岩岩先睇到 "n is a positive odd integer" 即係呢個唔可以用M.I. prove.. 唔好意思!!!