设计算法求1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)的值

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设计算法求1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)的值设计算法求1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)的值设计算法求1/(1*2)

设计算法求1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)的值
设计算法求1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)的值

设计算法求1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)的值
1/(1*2)+1/(2*3)+1/(3*4)+……+1/(99*100)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+.+(1/99-1/100)
=1-1/100
=99/100

1/(1*2)=1-1/2
1/(2*3)=1/2-1/3
1/(3*4)=1/3-1/4
……
1/(99*100)=1/99-1/100.
左边全部加起来=右边全部加起来。
右边=1-1/100

1/(1*2)=1/1-1/2
1/(2*3)=1/2-1/3
1/(3*4)=1/3-1/4
......
1/(99*100)=1/99-1/100
所以
原式=(1-1/2)+(1/2-1/3)+....+(1/99-1/100)
=1-1/100
=99/100

分数拆分公式:k/n(n+d)=k/d[1/n-1/(n+d)
上题答案为:1-1/2+1/2-1/3+1/3-1/4+1/4-……-1/99+1/99-1/100=1-1/100=99/100