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Đặt A = \(\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-\frac{1}{16}-\)...\(-\frac{1}{1024}\)
A= \(\frac{1}{2^1}-\frac{1}{2^2}-\frac{1}{2^3}-\frac{1}{2^4}-\)....\(-\frac{1}{2^{10}}\)
2A=\(\frac{1}{1}\)\(-\frac{1}{2^1}-\frac{1}{2^2}-\frac{1}{2^3}-\)...\(-\frac{1}{2^9}\)
2A-A=(\(\frac{1}{1}\)\(-\frac{1}{2^1}-\frac{1}{2^2}-\frac{1}{2^3}-\)...\(-\frac{1}{2^{10}}\)) \(-\)(\(\frac{1}{2^1}-\frac{1}{2^2}-\frac{1}{2^3}-\frac{1}{2^4}-\)..\(-\frac{1}{2^9}\))
A=\(1+\frac{1}{2^{10}}\)
A= \(\frac{1025}{1024}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(A=-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{1024}\)
=>\(2A=-2-1-\frac{1}{2}-\frac{1}{4}-...-\frac{1}{512}\)
=>\(2A-A=\left(-2-1-\frac{1}{2}-\frac{1}{4}-...-\frac{1}{512}\right)-\left(-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{1024}\right)\)
=>\(A=-2+\frac{1}{1024}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(A=-1-\dfrac{1}{2}-\dfrac{1}{4}-...-\dfrac{1}{1024}\)
\(\Leftrightarrow-A=1+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{1024}\)
\(\Leftrightarrow-\dfrac{1}{2}A=\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{11}}\)
\(\Leftrightarrow A\cdot\dfrac{1}{2}=\dfrac{1}{2^{11}}-1\)
hay \(A=\dfrac{2\cdot\left(1-2^{11}\right)}{2^{11}}=\dfrac{1-2^{11}}{2^{10}}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:
\(-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{1024}\)
\(=-1+\left(-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{1024}\right)\)
\(=-1+\left(-\frac{1}{2^1}-\frac{1}{2^2}-\frac{1}{2^3}-...-\frac{1}{2^{10}}\right)\)
\(=-1+\frac{-1023}{1024}\)
\(=-\frac{2047}{1024}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
gọi A=1/2+1/4+1/8+...+1/1024
2xA=1+1/2+1/4+.....+1/512
2xA-A=(1+1/2+1/4+....+1/512)-(1/2+1/4+1/8+...+1/1024)
A=1-1/1024
=1023/1024
vậy A=1023/1024
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(A=-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{1024}\)
\(A=-\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{1024}\right)\)
Đặt \(B=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{1024}\)
\(2B=2+1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{512}\)
\(2B-B=\left(2+1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{512}\right)-\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{1024}\right)\)
\(B=2-\frac{1}{1024}=\frac{2047}{1024}\)
=> \(A=-\frac{2047}{1024}\)
\(-1-\frac{1}{2}-\frac{1}{4}-.....-\frac{1}{1024}\)
\(=-\left(\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{10}}\right)\)
Giả sử A\(=-\left(\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{10}}\right)\)
=> - 2A=\(=1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^9}\)
\(\Rightarrow-2A+A\)\(=1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^9}\) + \(\left[-\left(\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{10}}\right)\right]\)
\(\Rightarrow-A=1-\frac{1}{2^{10}}\)
\(\Rightarrow A=\frac{1}{2^{10}-1}\)