1:3+1:6+1:10+1:15+1:21+1:28+1:36+1:45+1:55+1:66+1:78+1:91+1:105+1:120=?

1/3(1+1/2)+1/5(1/2+1/3)+1/7(1/3+1/4)+1/9(1/4+1/5)+1/11(1/5+1/6)+1/13(1/6+1/7)+1/15(1/7+1/8)
=1/3*3/2+1/5*5/6+1/7*7/12+1/9*9/20+1/11*11*30+1/13*13/42+1/15*15/56
=1/2+1/6+1/12+1/20+1/30+1/42+1/56
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8
=1-1/8
=7/8
关键是找这些数的规律,再加上裂项的技巧,问题就迎刃而解了!

原式＝1/（1×3）＋1/（2×3）＋1/（2×5）＋......＋1/（8×15）
=2[1/（2×3）＋1/（3×4）＋1/（4×5）＋.......1/（15×16）]
=2[(1/2-1/3)+(1/3-1/4)+(1/4-1/5).....+(1/15-1/16)]
=2{1/2+(-1/3+1/3)+(1/4+1/4)+.....+(-1/15+1/15)-1/16]
=2(1/2-1/16)
=1-7/8
=7/8

把1:3+1:6+1:10+1:15+1:21+1:28+1:36+1:45+1:55+1:66+1:78+1:91+1:105+1:120看成1/1*3+1/2*3+1/2*5+1/3*5+1/3*7+1/4*7+1/4*9+1/5*9+1/5*11+1/6*11+1/6*13+1/7*13+1/7*15+1/8*15；
把相连两项通分可得数列An＝（n＋1＋n)/n(n+1)（2n＋1）＝1/n（n＋1）n小于等于7，且为正数。
再利用求和公式可得7/8

1/1*3+1/2*3+1/2*5+1/3*5+1/3*7+1/4*7+1/4*9+1/5*9+1/5*11+1/6*11+1/6*13+1/7*13+1/7*15+1/8*15下面几步就是基本的简便计算了

1:a=1/a=2/n(n+1)=2*(1/n-1/(n+1))
1:3+1:6+1:10+1:15+1