巧算8/(3*3)+16/(15*15)+24/(35*35)+32/(63*63)+40/(99*99)+48/(143*143)

题目

巧算8/(3*3)+16/(15*15)+24/(35*35)+32/(63*63)+40/(99*99)+48/(143*143)

答案

先分析规律
15=3*5,35=5*7,63=7*9,99=9*11,143=11*13
16=5*5-3*3,24=7*7-5*5,32=9*9-7*7,40=11*11-9*9,48=13*13-11*11
再看原题
8/(3*3)=8/9=1-1/9=1-1/3*3
16/(15*15)=16/3*3*5*5=1/3*3-1/5*5
……
48/(143*143)=48/11*11*13*13=1/11*11-1/13*13
所以,
8/(3*3)+16/(15*15)+24/(35*35)+32/(63*63)+40/(99*99)+48/(143*143)
=1-1/3*3+1/3*3-1/5*5+……+1/11*11-1/13*13
=1-1/13*13
=1-1/169
=168/169