追夫记张张何何百度云:小学数学

解:(1)1/6 + 1/12 + 1/20 + 1/30 + 1/42
=1/2*3+1/3*4+1/4*5+1/5*6+1/6*7
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7
=1/2-1/7
=5/14
(2)1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90
=1/1*2+1/2*3+1/3*4+1/4*5+...+1/9*10
=1-1/2+1/2-1/3+1/3-1/4+...+1/9-1/10
=1-1/10
=9/10
(3)1/1*2 + 1/2*3 + 1/3*4 + 1/4*5 + 1/5*6......+ 1/98*99 + 1/99*100
=1-1/2+1/2-1/3+1/3-1/4+...+1/99-1/100
=1-1/100
=99/100

(1)1/6 + 1/12 + 1/20 + 1/30 + 1/42
=(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7
=1/2-1/7
=5/14
(2)1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90
=(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)+(1/8-1/9)+(1/9-1/10)
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10
=1/2-1/10
=2/5
(3)1/1*2 + 1/2*3 + 1/3*4 + 1/4*5 + 1/5*6+......+ 1/98*99 + 1/99*100
=(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+......+(1/98-1/99)+(1/99-1/100)
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+......+1/98-1/99+1/99-1/100
=1/2-1/100
=49/100

(1) =(1/2-1/3)+(1/3-1/4)+(1/5-1/6)+(1/6+1/7)
=1/2-1/3+1/3-1/4+1/5-1/6+1/6-1/7
=1/2-1/7
=5/14
下同
(2)=1-1/2+1/2-1/3+1/3-1/4+1/5-1/6+1/6+1/7+……1/9-1/10
=1-1/10
=9/10(3)=1-1/2+1/2-1/3+1/3-1/4+1/5-1/6+1/6+1/7+……1/99-1/100
=1-1/100
=99/100

听过裂项吧？就是 1/(n*(n+1))=1/n - 1/（n+1』

以上三题都可以用裂项轻松搞定。

如果对裂项不熟悉，先对第一题裂项看一看，结果很简单的。