1/2+1/6+1/12+1/20+1/30+...+1/9900怎么算

1/2+1/6+1/12+1/20+1/30+...+1/9900怎么算
1/2+1/6+1/12+1/20+1/30+...+1/9900怎么算

1/2+1/6+1/12+1/20+1/30+...+1/9900怎么算
每一项的分母可以写成n*(n+1)的形式,例如20=4*5,30=5*6,9900=99*100.那么每一项可以写成1/[n*(n+1)]的形式,又1/[n*(n+1)]=1/n-1/(n+1),所以原式=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+.+(1/99-1/100)=1-1/100=99/100

1/2=1/1-1/2,1/6=1/2-1/3,。。。,1/9900=1/99-1/100
所以相加=1-1/100=99/100

1/2+1/6+1/12+1/20+1/30+...+1/9900
=1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+……+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
=0.99

1/2=1-1/2
1/6=1/2-1/3
1/12=1/3-1/4
同理……所以原式=1-1/2+1/2-1/3+1/3+……-1/100=99/100

原式=(1-1/2)+(1/2-1/3)+(1/3-1/4)+…+(1/99-1/100)
=1-1/100
=99/100