1*N+2*(N-1)+3*(N-2)+...+N*1=1/6N(N+1)(N+2)

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1*N+2*(N-1)+3*(N-2)+...+N*1=1/6N(N+1)(N+2)
1*N+2*(N-1)+3*(N-2)+...+N*1=1/6N(N+1)(N+2)

1*N+2*(N-1)+3*(N-2)+...+N*1=1/6N(N+1)(N+2)
楼上再N+1证时有错误,正确的如下:
用数学归纳法:
当n=1时左边=1=右边
假设n=N时成立,
则n=N+1时:
1*(N+1)+2*(N+1-1)+3*(N+1-2)+...+N*(1+1)+(N+1)*1
=1*N+1 + 2*(N-1)+2 + 3*(N-2)+3+...+N*1+N + (N+1)
=1*N+2*(N-1)+3*(N-2)+...+N*1+(1+2+3+...+N+N+1)
=N(N+1)(N+2)/6+(N+1)*(N+2)/2
=[N(N+1)(N+2)+3*(N+1)*(N+2)]/6
=(N+1)(N+2)(N+3)/6
=(n)(n+1)(n+2)/6

第N项
1*N+2*(N-1)+3*(N-2)+...+N*1=1/6N(N+1)(N+2)
第N+1项
1*(N+1)+2*(N+1-1)+3*(N+1-2)+...+(N+1)*1
=1*N+1+2*(N-1)+2+3*(N-2)+3+...+N*1+1
=1*N+2*(N-1)+3*(N-2)+...+N*1+(1+2+3+...+n/2+n/2-1+...

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第N项
1*N+2*(N-1)+3*(N-2)+...+N*1=1/6N(N+1)(N+2)
第N+1项
1*(N+1)+2*(N+1-1)+3*(N+1-2)+...+(N+1)*1
=1*N+1+2*(N-1)+2+3*(N-2)+3+...+N*1+1
=1*N+2*(N-1)+3*(N-2)+...+N*1+(1+2+3+...+n/2+n/2-1+...+1)
=1/6N(N+1)(N+2)+(n+1)*(n+2)/2
=1/6(N+1)(N+2)(N+3)
1+2+3+...+n-1+n=n*(n+1)/2

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