歸納法問題?

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