Cho A=1+\(\frac{1}{3^2}\)+\(\frac{1}{3^4}\)+...+\(\frac{1}{3^{100}}\). Biết 8A=9-\(\frac{1}{3^{10}}\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
A = \(1+\frac{1}{3^2}+\frac{1}{3^4}+...+\frac{1}{3^{100}}\)
32A = \(9+1+\frac{1}{3^2}+...+\frac{1}{3^{98}}\)
9A - A = \(\left(9+1+\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)-\left(1+\frac{1}{3^2}+\frac{1}{3^4}+...+\frac{1}{3^{100}}\right)\)
8A = \(9-\frac{1}{3^{100}}=9-\frac{1}{3^n}\)
=> n = 100
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có 9A = 9 + 1 + \(\frac{1}{3^2}\) + \(\frac{1}{3^4}\) + ... + \(\frac{1}{3^{98}}\)
9A - A = 9 - \(\frac{1}{3^{100}}\) = 8A
Suy ra : 9 - \(\frac{1}{3^{100}}\) = 9 - \(\frac{1}{3^n}\)
\(\frac{1}{3^{100}}\) = \(\frac{1}{3^n}\)
Vậy n = 100
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=1+\frac{1}{3^2}+\frac{1}{3^4}+...+\frac{1}{3^{100}}\)
\(3^2A=9+1+\frac{1}{3^2}+...+\frac{1}{3^{98}}\)
\(9A-A=\left(9+1+\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)-\left(1+\frac{1}{3^2}+\frac{1}{3^4}+...+\frac{1}{3^{100}}\right)\)
\(8A=9-\frac{1}{3^{100}}\)
=> n = 100
![](https://rs.olm.vn/images/avt/0.png?1311)
de ot
9A-A=(\(9+1+\frac{1}{3^2}+...+\frac{1}{3^{99}}\))-\(\left(1+\frac{1}{3^2}+\frac{1}{3^4}+...+\frac{1}{3^{100}}\right)\)
8A=\(9-\frac{1}{3^{100}}\)
=>n=100