Question

用数学归纳法证明1~2-2~2+3~2-4~2+...+(2n-1)~2-(2n)~2=-n(2n+

用数学归纳法证明1~2-2~2+3~2-4~2+...+(2n-1)~2-(2n)~2=-n(2n+1)(n属于正整数)

Answer 1

1. n=1, 左边=1²-2²=1-4=-3
右边=-1*(2*1+1)=-3
左边=右边
2. 假设n=k, 1²-2²+3²-4²+...+(2k-1)²-(2k)²=-k(2k+1)
当 n=k+1时
1²-2²+3²-4²+....+(2k-1)²-(2k)²+(2k+1)²-(2k+2)²
=-k(2k+1)+(2k+1)²-(2k+2)²
=(2k+1)(2k+1-k)-4(k+1)²
=(k+1)(2k+1)-4(k+1)²
=(k+1)(2k+1-4k-4)
=(k+1)(-2k-3)
=-(k+1)(2(k+1)+1)
根据数学归纳法, 1²-2²+3²-4²+...+(2n-1)²-(2n)²=-n(2n+1)

Answer 2

两两结合
1²-2²+3²-4²+...+(2n-1)²-(2n)²=
(1-2)(1+2)+(3-4)(3+4)+...+(-1)(4n-1)
=-(3+7+11+15+...+4n-1)
=-n(4n+2)/2
=-n(2n+1)
数学归纳法，看jg_zhou的答案。

Answer 3

这是课本原题