1/1*3+1/2*4+1/3*5+1/4*6+.+1/18*20简便算法

1/1*3+1/2*4+1/3*5+1/4*6+.+1/18*20简便算法
1/1*3+1/2*4+1/3*5+1/4*6+.+1/18*20
简便算法

1/1*3+1/2*4+1/3*5+1/4*6+.+1/18*20简便算法
因为1/[n*(n+2)]=(1/2)*{(1/n)-[1/(n+2)]},
1/1*3+1/2*4+1/3*5+1/4*6+……+1/16*18+1/17*19+1/18*20
=(1/2)*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+……+1/16-1/18+1/17-1/19+1/18-1/20)
=(1/2)*(1+1/2-1/19-1/20)
=(1/2)*(531/380)
=531/760

1/1*3+1/2*4+1/3*5+1/4*6+......+1/18*20
=[1/(1*3)+1/(3*5)+...+1/(17*19)]+[1/(2*4)+1/(4*6)+...+1/(18*20)]
=1/2*(1-1/3+1/3-1/5+...+1/17-1/19)+1/2*(1/4-1/6+1/6-1/8+....+1/18-1/20)
=1/2*(1-1/19)+1/2(1/4-1/20)
=1/2*18/19+1/2*1/5
=9/19+1/10
=109/190

天哪

1/1*3+1/2*4+1/3*5+1/4*6+……+1/16*18+1/17*19+1/18*20
=(1/2)*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+……+1/16-1/18+1/17-1/19+1/18-1/20)
=(1/2)*(1+1/2-1/19-1/20)
=(1/2)*(531/380)
=531/760