Question

证明1²-2²+3²-4²+…+(2n-1)²-(2n)²=-n(2n+1).

Answer 1

证明:(1)当n=1时,左边=1²-2²=-3,?

右边=-1·(2×1+1)=-3,等式成立.?

(2)假设当n=k时,等式成立,即1²-2²+3²-4²+…+(2k-1)²-(2k)²=-k(2k+1),?

则当n=k+1时,1²-2²+…+(2k-1)²-(2k)²+(2k+1)²-(2k+2)²=-k(2k+1)+(2k+1)²-(2k+2)²=-(k-1)［2(k+1)

+1］.

由(1)(2)可知,对任何n∈N^*,等式成立.