1/2+5/6+11/12+…+109/110= 简便计算!

题目

1/2+5/6+11/12+…+109/110= 简便计算!

答案

1/2 + 5/6 + 11/12 +……+ 109/110
= (1 - 1/2) + (1 - 1/6) + (1 - 1/12) +……+ (1 - 1/110)
= [1 - 1/(1X2)] + [1 - 1/(2X3)] + [1 - 1/(3X4)] +……+ [1 - 1/(10X11)]
= 10 - [ 1/(1X2) + 1/(2X3) + 1/(3X4) +……+ 1/(10X11) ]
= 10 - [ (2 -1)/2 + (3 -2)/(2X3) + (4 -3)/(3X4) +……+ (11 -10)/(10X11) ]
= 10 - [ 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 +……+ 1/9 - 1/10 + 1/10 - 1/11 ]
= 10 - [ 1 - 1/11 ]
= 10 - 1 + 1/11
= 9+ 1/11