小学奥数拆分：1⼀(1*2*3*4)-1⼀(2*3*4*5)-1⼀(3*4*5*6)-……-1⼀(6*7*8*9)-1⼀(7*8*9*10)

看图片是+不是-啊？
先按+算。
首先
1/[n*(n+1)*(n+2)*(n+3)]=1/3 * {1/[n*(n+1)*(n+2)]-1/[(n+1)*(n+2)*(n+3)]}所以，原式=1/3[(1/1*2*3-1/2*3*4)+(1/2*3*4-1/3*4*5)...+(1/7*8*9-1/8*9*10)]=1/3(1/1*2*3-1/8*9*10)=119/2160

1/n(n+1)(n+2)(n+3)=1/2[1/n(n+3)+1/(n+1)(n+2)]剩下的就是对1/n(n+3) 和1/(n+1)(n+2)分组列项求和就是了1/n(n+3)=1/3[1/n-1/(n+3)] 1/(n+1)(n+2)=1/(n+1)-1/(n+2)剩下的自己加就是

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

题目有问题，正确的是：
5/(1*2*3*4)+7/(2*3*4*5)+9/(3*4*5*6)-……+15/(6*7*8*9)+17/(7*8*9*10)
=1/(1*3）-1/（2*4)+1/(2*4）-1/（3*5)+1/(3*5）-1/（4*6)+……+1/(6*8）-1/（7*9)+1/(7*9）-1/（8*10)
=1/（1*3）-1/（8*10）
=1/3-1/80
=77/240

1/(1*2*3*4)=1/2*[1/(1*4)-1/(2*3)]=1/6*(1/1-1/4)-1/2*(1/2-1/3)
1/(1*2*3*4)+1/(2*3*4*5)+1/(3*4*5*6)+……+1/(6*7*8*9)+1/(7*8*9*10)
=1/6*(1/1-1/4+1/2-1/5+1/3-1/6+...+1/6-1/9+1/7-1/10)-1/2*(1/2-1/3+1/3-1/4+1/4-1/5+...+1/7-1/8+1/8-1/9)
=1/6*(1/1+1/2+1/3-1/8-1/9-1/10)-1/2*(1/2-1/9)
=1/6*539/360-1/2*7/18
=119/2160