请教一道数学题1/2+3/4+7
S = 1/2 + 3/4 + 7/8 + 15/16 +………+ 2^(100-1)/2^100
设T = 1/2 + 1/4 + 1/8 + 1/16 +………+1/2^100
则T为等比数列，公比为1/2，所以
T = 1/2 + 1/4 + 1/8 + 1/16 +………+1/2^100
= (1/2)[1 - (1/2)^100]/(1 - 1/2)
= 1 - (1/2)^100
(S + T) = (1/2 + 1/2) + (3/4 + 1/4) + ……… + [2^(100-1)/2^100 + 1/2^100] = 1×100 = 100
S = (T + S) - ...全部
S = 1/2 + 3/4 + 7/8 + 15/16 +………+ 2^(100-1)/2^100
设T = 1/2 + 1/4 + 1/8 + 1/16 +………+1/2^100
则T为等比数列，公比为1/2，所以
T = 1/2 + 1/4 + 1/8 + 1/16 +………+1/2^100
= (1/2)[1 - (1/2)^100]/(1 - 1/2)
= 1 - (1/2)^100
(S + T) = (1/2 + 1/2) + (3/4 + 1/4) + ……… + [2^(100-1)/2^100 + 1/2^100] = 1×100 = 100
S = (T + S) - T = 100 - [1 - (1/2)^100] = 99 + (1/2)^100
即：1/2 + 3/4 + 7/8 + 15/16 +………+ 2^(100-1)/2^100 = 99 + (1/2)^100。
收起
