1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45+1/50+1/55=?用裂项法做

问题描述:

1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45+1/50+1/55=?
用裂项法做

设原式的和为S,再把原式乘以1/2得:
1/2S=1/2*(1/3+1/6+1/10+1/15+……+1 /55)
1/2S=1/6+1/12+1/20+1/30+1/42+……+1/110
1/2S=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+……+1/9-1/10+1/10-1/11
1/2S=1/2-1/11
S=9/11

设原式的和为S,再把原式乘以1/2得:
1/2S=1/2*(1/3+1/6+1/10+1/15+……+1/55)
=1/6+1/12+1/20+1/30+1/42+……+1/110
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+……+1/10-1/11
=1/2-1/11
=9/22
两边乘以2得:S=18/22