Question

1^3+2^3+3^3…+n^3的和的公式是什么

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Answer 1

S1=1^3=1=1^2
S2=1^3+2^3=S1+2^3=1+8=9=3^2
S3=S2+3^3=9+27=36=6^2
S4=S3+4^3=36+64=100=10^2
………………
Sn=1/4*n^2*(n+1)^2

Sn+1=Sn+(n+1)^3=1/4*n^2*(n+1)^2+(n+1)^3
=1/4*(n^2*(n+1)^2+4(n+1)^3)
=1/4*(n+1)^2*((n^2)+4(n+1))
=1/4*(n+1)^2*(n^2+4n+4)
=1/4(n+1)^2*(n+2)^2
数学归纳法证毕

Answer 2

S1=1^3=1=1^2
S2=1^3+2^3=S1+2^3=1+8=9=3^2
S3=S2+3^3=9+27=36=6^2
S4=S3+4^3=36+64=100=10^2
………………
Sn=1/4*n^2*(n+1)^2

Sn+1=Sn+(n+1)^3=1/4*n^2*(n+1)^2+(n+1)^3
=1/4*(n^2*(n+1)^2+4(n+1)^3)
=1/4*(n+1)^2*((n^2)+4(n+1))
=1/4*(n+1)^2*(n^2+4n+4)
=1/4(n+1)^2*(n+2)^2

Answer 3

[（1+n)*n/2]^2