1x2+2x3+3x4+.+n(n+1)=_____(n为自然数)
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1x2+2x3+3x4+.+n(n+1)=_____(n为自然数)
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
=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
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