Question

[Phamthithutrang]

giải bài toán sau

tính

S=1-1/2+1/4-1/8+1/16-...+1/1024

Answer 1

[Phùng Minh Quân]

Ta có : 

\(S=1-\frac{1}{2}+\frac{1}{4}-...+\frac{1}{1024}\)

\(S=1-\frac{1}{2}+\frac{1}{2^2}-...+\frac{1}{2^{10}}\)

\(\frac{1}{2}S=\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-...+\frac{1}{2^{11}}\)

\(S+\frac{1}{2}S=\left(1-\frac{1}{2}+\frac{1}{2^2}-...+\frac{1}{2^{10}}\right)+\left(\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-...+\frac{1}{2^{11}}\right)\)

\(\frac{3}{2}S=1-\frac{1}{2^{11}}\)

\(S=\frac{1-\frac{1}{2^{11}}}{\frac{3}{2}}\)

\(S=\frac{2-\frac{1}{2^{10}}}{3}\)

\(S=\frac{\frac{2^{11}-1}{2^{10}}}{3}\)

Vậy \(S=\frac{\frac{2^{11}-1}{2^{10}}}{3}\)

Chúc bạn học tốt ~ 

Answer 2

[Nguyễn Lưu Hà Phương]

Ta có:

2S = 2.(1-1/2+1/4-1/8+1/16-...+1/1024)

2S = 2/2-1+1/2-1/4+1/8-...+1/512

2S+S = ( 2/2-1+1/2-1/4+1/8-...+1/512)+(1-1/2+1/4-1/8+1/16-...+1/1024)

3S = 2 + 1/1024

3S = 2048/1024+1/1024

3S = 2049/1024

S = 2049/1024 : 3

S = 2049/1024.1/3

S = 683/1024