1/2+1/(2+3)+1/(2+3+4)+.+1/(2+4+5+...+200)=?能否简便?下面是我自己算的=1/2+2*[1/3*（1/（3-1）-1/(3+2)]+2*[1/3*(1/(4-1-1/(4+2)]+…+2*1/3*（1/（20-1）-1/（20+2）算到这就卡了请大侠用简便方法继续算…

Question

1/2+1/(2+3)+1/(2+3+4)+.+1/(2+4+5+...+200)=?能否简便?
下面是我自己算的
=1/2+2*[1/3*（1/（3-1）-1/(3+2)]+2*[1/3*(1/(4-1-1/(4+2)]+…+2*【1/3*（1/（20-1）-1/（20+2）】
算到这就卡了请大侠用简便方法继续算




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Answer 1

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.