Đặt
[TEX]A_1 = 1/2^1 + 1/2^2 + ... + 1/2^{100}
A_2 = 1/2^2 + 1/2^3 + ... + 1/2^{100}[/TEX]
[TEX]A_3 = 1/2^3 + 1/2^4 + ... + 1/2^{100}
....................................
...................................[/TEX]
[TEX]A_100 = 1/2^{100}
A = 1/2^1 + 2/2^2 + 3/2^3 + 4/2^4 + ... + 100/2^{100} =[/TEX]
[TEX]= (1/2^1+1/2^2 +...+ 1/2^{100}) + (1/2^2+1/2^3 +...+ 1/2^{100}) + (1/2^3+1/2^4 +...+ 1/2^{100}) + ... + (1/2^{100}) = A_1 + A_2 + A_3 + ... + A_{100}[/TEX]
[TEX]2^{101} A_1 = 2^{100} + 2^{99} + 2^{98} + ... + 2 (1)
2^{100} A_1 = 2^{99} + 2^{98} + 2^{97} + ... + 1 (2)[/TEX]
(2) trừ (1) [TEX]---> 2^{100 } A_1 = 2^{100} - 1 ---> A_1 = (2^{100} - 1) / 2^{100} = 1 - 1/2^{100} [/TEX]
Tương tự
[TEX]2^{101} A_2 = 2^{99} + 2^{98} + 2^{97} +...+ 2 (3)
2^100 A_2 = 2^98 + 2^97 + 2^96 +...+ 1 (4)[/TEX]
(4) trừ (3)[TEX] \Rightarrow 2^100 A_2 = 2^{99 } - 1 \Rightarrow A_2 = (2^{99} - 1) / 2^{100} = 1/2 - 1/2^{100}[/TEX]
Tương tự
[TEX]A_3 = 1/4 - 1/2^100 = 1/2^2 - 1/2^100
A_4 = 1/2^3 - 1/2^{100}[/TEX]
[TEX]..................................
.................................
A_100 = 1/2^{99} - 1/2^{100}[/TEX]
Vậy [TEX]A = A_1 + A_2 + A_3 +...+ A_100 = (1 + 1/2 + 1/2^2 + ... + 1/2^{99}) - 100/2^{100}[/TEX]
[TEX]= 2 A_1 - 100/2^{100} = 2 - 2/2^{100} - 100/2^{100} = 2 - 51/2^{99}[/TEX]
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn