Question

计算1/2+1/(2+3)+1/(2+3+4)+....+1/(2+3+4+...+200)

这是等差数列，那章的。。

Answer 1

1/2+1/(2+3)+1/(2+3+4)+......+1/(2+3+4+5+...200)
=1/2+1/[(2+3)*2/2]+1/[(2+4)*3/2]+……+1/[(2+200)*199/2]
=1/2+2/(2*5)+2/(3*6)+2/(4*7)+……+2/(199*202)
=1/2+2/3[(1/2-1/5)+(1/3-1/6)+(1/4-1/7)+……+(1/199-1/202)]
=1/2+2/3[1/2-1/5+1/3-1/6+1/4-1/7+1/5-1/8+……+1/199-1/202]
=1/2+2/3[1/2+1/3+1/4+1/5+1/6+…………+1/198+1/199
-1/5-1/6-…………-1/198-1/199-1/200-1/201-1/202]
=1/2+2/3[(1/2+1/3+1/4-1/200-1/201-1/202)
=1/2+2/3[13/12-（20301+20200+20100）/4060200]
=1/2+2/3[13/12-60601/4060200]
=1/2+2/3*4337949/4060200
=1/2+1445983/2030100
=(1015050+1445983)/2030100
=2461033/2030100

对！运算过程中涉及到等差数列的概念。

就是计算题中各个分母时，可以看到他们各自是有2到n的等差数列组成，那么它们的和就是利用等差数列求和的公式计算的。就是（首项+尾项）×项数÷2。

就是这一步
=1/2+1/[(2+3)*2/2]+1/[(2+4)*3/2]+……+1/[(2+200)*199/2]

接着又利用了：1/n-1/(n+k)=1/k[1/n(n+k)]

即这几步
=1/2+2/(2*5)+2/(3*6)+2/(4*7)+……+2/(199*202)
=1/2+2/3[(1/2-1/5)+(1/3-1/6)+(1/4-1/7)+……+(1/199-1/202)]
=1/2+2/3[1/2-1/5+1/3-1/6+1/4-1/7+1/5-1/8+……+1/199-1/202]
=1/2+2/3[1/2+1/3+1/4+1/5+1/6+…………+1/198+1/199
-1/5-1/6-…………-1/198-1/199-1/200-1/201-1/202]

最后符号相反的相同数字抵消（从1/5、1/6……1/199），只剩下后面给出的各项，经过运算得出最后结果

Answer 2

1/2+1/(2+3)+1/(2+3+4)+…+1/(2+3+4+…+200)
=1/(1*4/2)+1/(2*5/2)+1/(3*6/2)+…+1/(199*202/2)
=2[1/(1*4)+1/(2*5)+1/(3*6)+…+1/(199*202)]
=(2/3)(1-1/4+1/2-1/5+1/3-1/6+1/4-1/7+…1/197-1/200+1/198-1/201+1/199-1/202)
=(2/3)(1+1/2+1/3-1/200-1/201-1/202)
从你下边的计算上看好像不是加到200，而是加到20。
1/2+1/(2+3)+1/(2+3+4)+…+1/(2+3+4+…+20)
=1/(1*4/2)+1/(2*5/2)+1/(3*6/2)+…+1/(19*22/2)
=2[1/(1*4)+1/(2*5)+1/(3*6)+…+1/(19*22)]
=(2/3)(1-1/4+1/2-1/5+1/3-1/6+1/4-1/7+…1/17-1/20+1/18-1/21+1/19-1/22)
=(2/3)(1+1/2+1/3-1/20-1/21-1/22)
结果你自己算一下吧。

Answer 3

1/2+1/(2+3)+1/(2+3+4)+......+1/(2+3+4+5+...200)
=1/2+1/[2*(2+3)/2]+1/[3*(2+4)/2]+……+1/[199*(2+200)/2]
=1/2+2/(2*5)+2/(3*6)+2/(4*7)+……+2/(199*202)
=1/2+2/3(1/2-1/5)+2/3(1/3-1/6)+2/3(1/4-1/7)+……+2/3(1/199-1/202)
=1/2+2/3(1/2-1/5+1/3-1/6+1/4-1/7+1/5-1/8+……+1/199-1/202)
=1/2+2/3(1/2+1/3+1/4-1/200-1/201-1/202)
=1/2+2/3(13/12-30301/2030100)
=1/2+2/3*2168974/2030100
=1/2+2168974/3045150
=3691549/3045150

Answer 4

我也不懂啊