1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90==

1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90==
1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90
=
=

1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90==
1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90
＝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)+(1/8-1/9)+(1/9-1/10)
=1/2+1/2-1/10=9/10

1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8+1/8*9+1/9*10
1/1*2=1/1-1/2
1/2*3=1/2-1/3
后面的同理
自己算结果吧

=(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/8-1/9)+(1/9-1/10)=1-1/10=0.9

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+1/8-1/9+1/9-1/10
=1-1/10
=9/10

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+1/8-1/9+1/9-1/10
=1-1/10
=9/10

1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90
=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)+(1/8-1/9)+(1/9-1/10)
=1/2+1/2-1/10
=9/10

=1/n*(n+1)
=1/n*1/(n+1)
=1/1*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+1/8*1/9+1/9*1/10
=1/1*1/2+1/3*(1/2+1/4)+1/5*(1/4+1/6)+1/7*(1/6+1/8)+1/9*(1/8+1/10)
=1/2+1/4+1/12+1/24+1/40
=9/10

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

=1/2(1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45)
=1/2[1+1/3(1+1/2)+1/5(1/2+1/3)+1/7(1/3+1/4)+1/9(1/5+1/4)]
=1/2(1+1/2+1/6+1/12+1/20)
=1/2(1+3/4+1/20)
=1/2(1+16/20)
=1/2(1+4/5)
=1/...

全部展开

=1/2(1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45)
=1/2[1+1/3(1+1/2)+1/5(1/2+1/3)+1/7(1/3+1/4)+1/9(1/5+1/4)]
=1/2(1+1/2+1/6+1/12+1/20)
=1/2(1+3/4+1/20)
=1/2(1+16/20)
=1/2(1+4/5)
=1/2*(9/5)
=9/10 唉。这些数字符号打得我好累啊，你看看，应该对的吧……
像类似的题目，看起来复杂的都会有规律的，比如这些分数形式的运算，都可以从分解的角度去考虑

收起

=1/2(1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45)
=1/2[1+1/3(1+1/2)+1/5(1/2+1/3)+1/7(1/3+1/4)+1/9(1/5+1/4)]
=1/2(1+1/2+1/6+1/12+1/20)
=1/2(1+3/4+1/20)
=1/2(1+16/20)
=1/2(1+4/5)
=1/2*(9/5)
=9/10