Question

[nguyen gia huy]

Tính nhanh:A = 1/2 + 1/2² + 1/2³ + ............................... + 1/2¹⁰

Answer 1

[Le Thi Khanh Huyen]

Ta có:

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

\(\Rightarrow2A=1+\frac{1}{2}+...+\frac{1}{2^9}\)

Lấy \(2A-A\), ta có:

\(2A-A=A=\left(1+\frac{1}{2}+...+\frac{1}{2^9}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\right)\)

\(=1+\frac{1}{2}+...+\frac{1}{2^9}-\frac{1}{2}-\frac{1}{2^2}-...-\frac{1}{2^{10}}\)

\(=\left(1-\frac{1}{2^{10}}\right)+\left(\frac{1}{2}-\frac{1}{2}\right)+...+\left(\frac{1}{2^9}-\frac{1}{2^9}\right)\)

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

\(=1-\frac{1}{1024}\)

\(=\frac{1023}{1024}\)

Vậy \(A=\frac{1023}{1024}\)