Mathematical Induction

Let P(n) be x^n-y^n is divisible by x-y for all natural numbers n

When n = 1,

x^1-y^1
=x-y
which is divisible by x-y
So P(1) is true.

Assume P(k) is true, i.e. x^k-y^k is divisible by x-y
so x^k-y^k = (x-y)m where m is a positive integer. ... (*)

When n = k+1,
x^(k+1)-y^(k+1)
=x(x^k)-y(y^k)
=x(x^k-y^k)+xy^k-y(y^k)
=x(x-y)m+y^k(x-y) ............. (use (*))
=(x-y)(mx+y^k)
which is divisble by x-y
So P(k+1) is true.

by Mathemtical Induction, x^n-y^n is divisible by x-y for all natural numbers n

2006-11-09 19:01:05 補充：
小小提示：最緊要是在 P(k+1) 時怎樣可以運用 P(k)。其實任何有關 a isdivisible by b 的 MI 題目，將其轉換為 a = bm (for +ve integer m) 就可以較易在 P(k+1) 時運用。