Question

观察下列一组算式3^2-1^2=8=8*1,5^2-3^2=16=8*2,7^2-5^2=24=8*3

Answer 1

3^2-1^2=8=8*1,

5^2-3^2=16=8*2,

7^2-5^2=24=8*3

.

(2n+1)^2-(2n-1)^2=4n*2=8*n

规律:任意两个奇数的平方差等于8的倍数

1+2+3……+n

=(8*1+8*2+8*3……+8*n)/8

=[3^2-1^2+5^2-3^2+7^2-5^2+...+(2n+1)^2-(2n-1)^2]/8

=[(2n+1)^2-1^2]/8

=[(2n+1+1)(2n+1-1)]/8

=4n(n+1)/8

=n(n+1)/2