Question

计算1\1×2×3×4+1\2×3×4×5+...+1\7×8×9×10 快快！

要巧算，要过程，详细！在今天12点半以前回答出来的（并肯定对滴）加财富30！！！！！！！！

Answer 1

这是一个1/n(n+1)(n+2)(n+3)的形式的和
n(n+3)=n^2+3n;
(n+1)(n+2)=n^2+3n+1;
(n+1)(n+2)-n(n+3)=1
则原来的式子可以写成：
1/n(n+1)(n+2)(n+3)
=[(n+1)(n+2)-n(n+3)]/[n(n+1)(n+2)(n+3)]
=1/[n(n+3)]-1/[(n+2)(n+1)]
=1/2[1/n-1/(n+3)+1/(n+2)-1/(n+1)]

∴原式=1/2(1-1/2+1/3-1/4)+1/2(1/2-1/3+1/4-1/5)+……+1/2(1/7-1/8+1/9-1/10)
=1/2[(1-1/2+1/3-1/4)+(1/2-1/3+1/4-1/5)+……+(1/7-1/8+1/9-1/10)]
=1/2(1-1/2+1/3-1/4+1/2-1/3+1/4-1/5+……+1/7-1/8+1/9-1/10)
=1/2(1-1/10)
=1/2*9/10
=9/20

此题必对，ohla

Answer 2

1/n(n+1)(n+2)(n+3)=[1/n(n+1)(n+2)-1/(n+1)(n+2)(n+3)]/3
1/1×2×3×4+1/2×3×4×5+...+1/7×8×9×10
=(1/1*2*3-1/2*3*4+1/2*3*4-1/3*4*5+...+1/7*8*9-1/8*9*10)/3
=(1/1*2*3-1/8*9*10)/3
=(1/6-1/720)/3
=119/2160

追问

稍等，等我知道正确答案后才能给你加财富值！
等6小时 呵呵