51问答网 > (1+2分之1)乘(1-2分之1)+(1+3分之1)乘(1-3分之1)+...(1+99分之1)乘(1-99分之1)

(1+2分之1)乘(1-2分之1)+(1+3分之1)乘(1-3分之1)+...(1+99分之1)乘(1-99分之1)

2025-01-07 02:24:10

推荐回答（2个）

回答1：

(1+1/2)*(1-1/2)*(1+1/3)*(1-1/3)*...*(1+1/99)*(1-1/99)
=(1+1/2)*(1+1/3)*...(1+1/99)*(1-1/2)*(1-1/3)*...*(1-1/99)(乘法交换律)
=3/2*4/3*5/4*6/5*7/6...97/96*98/97*99/98*100/99*1/2*2/3*3/4**4/5*5/6...96/97*97/98*98/99
=1/2*100*1/99
=50/99

回答2：

(1+1/2)(1-1/2) + (1+1/3)(1-1/3) + ... + (1+1/99)(1-1/99)
= (1-1/4) + (1-1/9) + ... + (1-1/99²)
= 98 - (1/4 + 1/9 + ... + 1/99²)