Question

1*2+2*3+3*4+...........19*（19+1）=？？通用公式是？？怎么计算出来给个计算过程

Answer 1

1*2+2*3+……+n*(n+1)
=1^2+1+2^2+2+……+n^2+n
=(1^2+2^2+……+n^2)+(1+2+……+n)
=n(n+1)(2n+1)/6+n(n+1)/2
=n(n+1)(n+2)/3

注:1^2+2^2+……+n^2=n(n+1)(2n+1)/6

是公式 不知道怎么推倒的接着问 我来答

Answer 2

您好!
解:
设 S = 1*2+2*3+3*4+……+19*20,
则
3S
= 1*2*3+2*3*(4-1)+3*4*(5-2)+……+19*20*(21-18)
= 1*2*3-1*2*3+2*3*4-2*3*4+3*4*5-3*4*5+……+18*19*20-18*19*20+19*20*21
= 19*20*21
= 20^3-20
= 7980.
所以 S = 2660.

Answer 3

1*2+2*3+……+n*(n+1)=1^2+1+2^2+2+……+n^2+n
=(1^2+2^2+……+n^2)+(1+2+……+n)
=n(n+1)(2n+1)/6+n(n+1)/2
=n(n+1)(n+2)/3

Answer 4

原式=1+2+3…+N+1^2+2^2+…N^2=N(N+1)/2+N(N+1)(2N+1)/6=2660