1+3又1/6+5又1/12+7又1/20+9又1/30+11又1/42+13又1/56+15又1/72+17又1/90的简便算法

1+3又1/6+5又1/12+7又1/20+9又1/30+11又1/42+13又1/56+15又1/72+17又1/90的简便算法
1+3又1/6+5又1/12+7又1/20+9又1/30+11又1/42+13又1/56+15又1/72+17又1/90的简便算法

1+3又1/6+5又1/12+7又1/20+9又1/30+11又1/42+13又1/56+15又1/72+17又1/90的简便算法
1+3又1/6+5又1/12+7又1/20+9又1/30+11又1/42+13又1/56+15又1/72+17又1/90
=(1+3+5+...+17)+1/2*3+1/3*4+...+1/9*10
=（1+3+5+...+17）+[(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...+(1/9-1/10)]
=81+(1/2-1/10)
=81又2/5

原式=(1+3+5+7+9+11)+(1/6+1/12+1/20+1/30+1/42)
=12*3+[(1/2)-(1/3)+(1/3)-(1/4)+(1/4)-(1/5)+(1/5)-(1/6)+(1/6)- (1/7)]
=36+(1/2)-(1/7)
=36+5/14=36/5/14(即为36又14分之5)

原式
=1+[3+1/(2*3)]+[5+1/(3*4)]+[7+1/(4*5)]+[9+1/(5*6)]+[11+1/(6*7)]+[13+1/(7*8)]+[15+1/(8*9)]+[17+1/(9*10)]
=(1+3+5+7+9+11+13+15+17)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)+...

全部展开

原式
=1+[3+1/(2*3)]+[5+1/(3*4)]+[7+1/(4*5)]+[9+1/(5*6)]+[11+1/(6*7)]+[13+1/(7*8)]+[15+1/(8*9)]+[17+1/(9*10)]
=(1+3+5+7+9+11+13+15+17)+(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+3+5+7+9+11+13+15+17+1/2-1/10
=(3+17)+(5+15)+(7+13)+(9+11)+1+1/2-1/10
=80+3/2-1/10
=81.4

收起

原式=1+（3+17）+（5+15）+（9+11）+（13+7）+(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+3+5+7+9+11+13+15+17+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=81又2/5