Question

Rút gọn biểu thức:

A = 1 + \(\frac{1}{2}\)+\(\frac{1}{2^2}\)+ \(\frac{1}{2^3}\)+ .......+\(\frac{1}{2^{2012}}\)

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Answer 1

\(A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\)

\(2A=2+1+\frac{1}{2}+...+\frac{1}{2^{2011}}\)

\(2A-A=(2+1+\frac{1}{2}+...+\frac{1}{2^{2011}})-(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2012}})\)

\(A=2-\frac{1}{2^{2012}}\)

Vậy A = \(2-\frac{1}{2^{2012}}\)

~Chúc bạn học tốt~

Answer 2

Xét\(2A=2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}\)

Lấy 2A - A Ta được

\(A=2-\frac{1}{2^{2012}}\)

Answer 3

\(A=1+\frac{1}{2}+...+\frac{1}{2^{2012}}\)

\(2A=2+1+\frac{1}{2}+...+\frac{1}{2^{2011}}\)

\(2A-A=\left(2+1+\frac{1}{2}+...+\frac{1}{2^{2011}}\right)-\left(1+\frac{1}{2}+...+\frac{1}{2^{2012}}\right)\)

=> \(A=2-\frac{1}{2^{2012}}\)

Answer 4

Trả lời

\(A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{...1}{2^{2012}}\)

Đặt \(B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+..+\frac{1}{2^{2012}}\)

\(\Leftrightarrow2B=\frac{2}{2}+\frac{2}{2^2}+\frac{2}{2^3}+...+\frac{2}{2^{2012}}\)

\(\Leftrightarrow2B=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}\)

\(\Leftrightarrow2B-B=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{...1}{2^{2012}}\right)\)

\(\Leftrightarrow B=1-\frac{1}{2^{2012}}\)

\(\Leftrightarrow A=1+\left(1-\frac{1}{2^{2012}}\right)\)

\(\Leftrightarrow A=2-\frac{1}{2^{2012}}\)

Vậy \(A=2-\frac{1}{2^{2012}}\)

Answer 5

Ta có : 2A = 2 (1 + 1/2 + 1/2^2 + 1/2^3 +......+1/2^2012)

                 = 2 + 1 + 1/2 + 1/2^2 + .....+ 1/2^2011

=> A = 2A -A = 2 - 1/2^2012