1/3+1/15+1/35+1/63+1/99+1/143+1/195简便

1/3+1/15+1/35+1/63+1/99+1/143+1/195简便
1/3+1/15+1/35+1/63+1/99+1/143+1/195简便

1/3+1/15+1/35+1/63+1/99+1/143+1/195简便
1/3=1/2(1-1/3)
1/15=1/2(1/3-1/5)
1/35=1/2(1/5-1/7)
1/63=1/2(1/7-1/9)
.
.
1/195=1/2(1/13-1/15)
1/3+1/15+1/35+1/63+1/99+1/143+1/195
=1/2(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+...-1/13+1/13-1/15)
=1/2(1-1/15)
=1/2*14/15
=7/15

解 ： 1/3 = 1/1 x 3 = (1-1/3)/2
15 = 1/3 x 5 = (1/3 - 1/5)/2
35 = 1/5 x 7 ....
195 = 1/13 x 15 = (1/13 - 1/15)/2
所以 1/3+1/15+1/35+1/63+1/99+1/143+1/195
= (1-1/3+1/3-1/5+1/5-1/7+...+1/13-1/15)/2
= 7/15

1/3+1/15+1/35+1/63+1/99+1/143+1/195
=1/1*3+1/3*5+1/5*7+1/7*9+1/9*11+1/11*13+1/13*15
=1/2*(1-1/3)+1/2*(1/3-1/5)+1/2*(1/5-1/7)+1/2*(1/7-1/9)+1/2*(1/9-1/11)+1/2*(1/11-1/13)+1/2*(1/13-1/15)
=1/2*(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+1/9-1/11+1/11-1/13+1/13-1/15)
=1/2*(1-1/15)
=1/2*14/15
=7/15

1/（1x3）+1/（3x5）+1/（5x7）+1/（7x9）+1/（9x11）+1/（11x13）+1/（13x15）+1/（15x17）
=(1-2/3)+(2/3-3/5)+(3/5-4/7)+(4/7-5/9)+(5/9-6/11)+(6/11-7/13)+(7/13-8/15)+ (8/15-9/17)＝1-9/17＝8/17