1^3+2^3+...+n^3=?怎么推导?

2024-11-19 03:40:13
推荐回答(3个)
回答1:

1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6

利用立方差公式
n^3-(n-1)^3=1*[n^2+(n-1)^2+n(n-1)]
=n^2+(n-1)^2+n^2-n
=2*n^2+(n-1)^2-n

2^3-1^3=2*2^2+1^2-2
3^3-2^3=2*3^2+2^2-3
4^3-3^3=2*4^2+3^2-4
......
n^3-(n-1)^3=2*n^2+(n-1)^2-n

各等式全相加
n^3-1^3=2*(2^2+3^2+...+n^2)+[1^2+2^2+...+(n-1)^2]-(2+3+4+...+n)

n^3-1=2*(1^2+2^2+3^2+...+n^2)-2+[1^2+2^2+...+(n-1)^2+n^2]-n^2-(2+3+4+...+n)

n^3-1=3*(1^2+2^2+3^2+...+n^2)-2-n^2-(1+2+3+...+n)+1

n^3-1=3(1^2+2^2+...+n^2)-1-n^2-n(n+1)/2

3(1^2+2^2+...+n^2)=n^3+n^2+n(n+1)/2=(n/2)(2n^2+2n+n+1)
=(n/2)(n+1)(2n+1)

1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6

1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2

(n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2]
=(2n^2+2n+1)(2n+1)
=4n^3+6n^2+4n+1

2^4-1^4=4*1^3+6*1^2+4*1+1
3^4-2^4=4*2^3+6*2^2+4*2+1
4^4-3^4=4*3^3+6*3^2+4*3+1
......
(n+1)^4-n^4=4*n^3+6*n^2+4*n+1

各式相加有
(n+1)^4-1=4*(1^3+2^3+3^3...+n^3)+6*(1^2+2^2+...+n^2)+4*(1+2+3+...+n)+n

4*(1^3+2^3+3^3+...+n^3)=(n+1)^4-1+6*[n(n+1)(2n+1)/6]+4*[(1+n)n/2]+n
=[n(n+1)]^2

1^3+2^3+...+n^3=[n(n+1)/2]^2

回答2:

(n+1)^4-n^4=4n^3+6n^2+4n+1
n^4-(n-1)^4=4(n-1)^3+6(n-1)^2+4(n-1)+1
……
2^4-1^4=4*1^3+6*1^2+4*1+1
都加起来,左边正负抵消
(n+1)^4-1^4=4*(1^3+2^3+……+n^3)+6*(1^2+2^2+……+n^2)+4*(1+2+……+n)+n*1

1^3+2^3+……+n^3=[(n+1)^4-1^4-6*(1^2+2^2+……+n^2)-4*(1+2+……+n)-n]/4

因为1^2+2^2+……+n^2=n(n+1)(2n+1)/6
1+2+……+n=n(n+1)/2
代入整理
得1^3+2^3+……+n^3=n^2(n+1)^2/4

注意:
附:1^2+2^2+……+n^2=n(n+1)(2n+1)/6 的推导同上
2^3-1^3=3*1^2+3*1+1
3^3-2^3=3*2^2+3*2+1
................
(n+1)^3-n^3=3*n^2+3*n+1

以上各式叠加:

(n+1)^3-1^3=3(1^2+2^2+.......+n^2)+3*(1+2+...+n)+n

(1^2+2^2+.......+n^2)=(N-1)N(2N-1)/6

回答3:

(n+1)^4-n^4=4n^3+6n^2+4n+1
n^4-(n-1)^4=4(n-1)^3+6(n-1)^2+4(n-1)+1
……
2^4-1^4=4*1^3+6*1^2+4*1+1
都加起来,左边正负抵消
(n+1)^4-1^4=4*(1^3+2^3+……+n^3)+6*(1^2+2^2+……+n^2)+4*(1+2+……+n)+n*1

1^3+2^3+……+n^3=[(n+1)^4-1^4-6*(1^2+2^2+……+n^2)-4*(1+2+……+n)-n]/4

因为1^2+2^2+……+n^2=n(n+1)(2n+1)/6
1+2+……+n=n(n+1)/2
代入整理
得1^3+2^3+……+n^3=n^2(n+1)^2/4

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