1*2*3+2*3*4+……+n（n+1）（n+2）=?
答案:6 悬赏:30 手机版
解决时间 2021-12-16 11:31
- 提问者网友：人走茶凉
- 2021-12-16 06:28
1*2*3+2*3*4+……+n（n+1）（n+2）=?
最佳答案
- 五星知识达人网友：只是一只团子
- 2021-12-16 07:13
n(n+1)(n+2)=n^3+3n^2+2n
1^3+2^3+......+n^3 =[n(n+1)/2]^2
1^2+2^2+.....+n^2=n(n+1)(2n+1)/6
1+2+3+...+n=n(n+1)/2
1*2*3+2*3*4+n(n+1)(n+2)=[n(n+1)/2]^2 +n(n+1)(2n+1)/2+n(n+1)
=n(n+1)/2(n(n+1)/2+2n+3)=n(n+1)(n^2+5n+6)/4=n(n+1)(n+2)(n+3)/4
1^3+2^3+......+n^3 =[n(n+1)/2]^2
1^2+2^2+.....+n^2=n(n+1)(2n+1)/6
1+2+3+...+n=n(n+1)/2
1*2*3+2*3*4+n(n+1)(n+2)=[n(n+1)/2]^2 +n(n+1)(2n+1)/2+n(n+1)
=n(n+1)/2(n(n+1)/2+2n+3)=n(n+1)(n^2+5n+6)/4=n(n+1)(n+2)(n+3)/4
全部回答
- 1楼网友：桃子不奈甜
- 2021-12-16 10:10
什么玩意，先乘后加？？？
- 2楼网友：抚你发端
- 2021-12-16 09:38
有： ① 2*(1+2+3+4+...+N)=N(N+1) ;
② 3*(1^2+2^2+3^2+...+N^2)=3*[N(N+1)(2N+1)/6] ;
③ 1^3+2^3+3^3+...+N^3=[N(N+1)/2]^2 ;
由①+②+③得：
1*2*3+2*3*4+······+N（N+1)(N+2)
=N(N+1)(N+2)(N+3)/4
- 3楼网友：心本逍遥
- 2021-12-16 08:36
因为(n-1)*n*(n+1)=n^3-n 所以 1*2*3+2*3*4…+99*100*101 =2^3+3^3+4^3+…+100^3-2-3-4-…- 100 =1^3+2^3+3^3+4^3+…+100^3-1-2-3-4-…- 100 =(100*101/2)^2-5050 =25497450 （ 因为^3+2^3+3^3+4^3+…+n^3=(n*(n+1)/2)^2)
- 4楼网友：一杯清酒当人生
- 2021-12-16 08:19
n(n+1)(n+2)
=n(n²+3n+2)
=n³+3n²+n
∴1×2×3+2×3×4+…n(n+1)(n+2)
=(1³+3*1²+1)+(2³+3*2²+2)+……+(n³+3n²+n)
=(1³+2³+……+n³)+3(1²+2²+……n²)+(1+2+……n)
=[n(n+1)/2]^2+3[n(n+1)(2n+1)/6]+n(n+1)/2
=n²(n+1)²/4+n(n+1)(2n+1)/2+n(n+1)/2
=n(n+1)[n(n+1)+2(2n+1)+2]/4
=n(n+1)(n²+n+4n+2+2)/4
=n(n+1)(n²+5n+4)/4
=n(n+1)(n+1)(n+4)/4
=n(n+1)²(n+4)/4
- 5楼网友：尘事揪人心
- 2021-12-16 07:18
1×2×3+2×3×4+3×4×5+......+n(n+1)(n+2)
=1/4【1×2×3×4-0×1×2×3】+1/4【2×3×4×5-1×2×3×4】+1/4【3×4×5×6-2×3×4×5】+......+
1/4【n(n+1)(n+2)（n+3)-(n-1)n(n+1)(n+2)】
=1/4n(n+1)(n+2)(n+3)
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