標題:
Mathematical Induction
發問:
Prove by Mathematical Induction that x^n-y^n is divisible by x-y for all natural numbers n
最佳解答:
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。
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