Question

1x2+2x3+3x4+.+n(n+1)=_____(n为自然数）

Answer 1

1*2+2*3+.+N（N+1）

=1*2+2*3+.+(N^2+N)

=(1+2+3+...+N)+(1^2+2^2+...+N^2)

=[N(N+1)/2]+[N(N+1)(2N+1)/6]

=N(N+1)(N+2)/3

1*2*3+2*3*4+.+N(N+1)*(N+2)

=1*2*3+2*3*4+.+(N^3+3N^2+2N)

=(1^3+2^3+...+N^3)+3(1^2+2^2+...+N^2)+2(1+2+...+N)

=[N(N+1)/2]^2+[N(N+1)(2N+1)/2]+N(N+1)

=(N+1)(N^3+3N^2+6N)/4