https://i.imgur.com/IXUGuRU.jpg
\(\left(3^n+\frac{1}{3^n}\right)^2=3^{2n}+\frac{1}{3^{2n}}+2\)
\(\Rightarrow S=2n+\left(3^2+3^4+...+3^{2n}\right)+\left(\frac{1}{3^2}+\frac{1}{3^4}+...+\frac{1}{3^{2n}}\right)\)
\(=2n+\left(9+9^2+...+9^n\right)+\left(\frac{1}{9}+\frac{1}{9^2}+...+\frac{1}{9^n}\right)\)
Ngoặc đầu tiên là tổng CSN có \(u_1=9;q=9\), ngoặc thứ 2 là tổng CSN có \(u_1=\frac{1}{9};q=\frac{1}{9}\)
\(\Rightarrow S=2n+\frac{9\left(9^n-1\right)}{8}+\frac{1}{9}.\frac{1-\frac{1}{9^n}}{1-\frac{1}{9}}=2n+\frac{1}{8}.9^{n+1}-\frac{1}{8.9^n}\)
