1/1*2*3+1/2*3*4+1/3*4*5+.1/99*100*101的方法步骤
问题描述：
1/1*2*3+1/2*3*4+1/3*4*5+.1/99*100*101的方法步骤
说清楚点.
答
考察一般项：
1/[n(n+1)(n+2)]
=(1/2)[1/n -1/(n+2)] -[1/(n+1)-1/(n+2)]
=(1/2)(1/n) -1/(n+1) +(1/2)[1/(n+2)]
=(1/2)[1/n -2/(n+1)+1/(n+2)]
=(1/2){[1/n -1/(n+1)]-[1/(n+1)-1/(n+2)] }
1/(1×2×3)+1/(2×3×4)+1/(3×4×5)+...+1/(99×100×101)
=(1/2)[(1/1-1/2)-(1/2-1/3)+(1/2-1/3)-(1/3-1/4)+...+(1/99-1/100)-(1/100-1/101)]
=(1/2)(1/1-1/2-1/100+1/101)
=5049/20200什么原理？裂项法。