標題:
歸納法問題?
發問:
証明：x^(2y) - y^(2y)能被x+y整除 (用歸納法)
最佳解答:
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我想，你的問題應該是 X^2n - y ^ 2n is divisible by x+y for all natural number n For n=1 x^2 - y^2 = (x+y)(x-y)..... obviously divisible by x+y Assume n=k it is true, that is Assumption１：x^2k - y^2k is also divisible by x+y Assumption2： x^2k-2 - y^2k-2 also divisible by x+y For n=k+1 x^2(k+1) - y^2(k+1) = x^2 ?x^2k - y^2?y^2k = x^2 ?x^2k - (x^2? y^2k - x^2? y^2k ) - y^2?y^2k+ (y^2?x^2k - y^2?x^2k) = x^2(x^2k-y^2k) + y^2(x^2k-y^2k) - (xy)^2[x^2k-2 - y^2k-2] So, 3 part of them all divisible by x+y first part and second part same, use assumption1: x^2k-y^2k, third part use assumption2 : x^2k-2 - y^2k-2 there for, by M.I. x^2n - y^2n is divisible by x+y for all n >=1 2007-05-18 08:53:05 補充： 唔明左問我
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