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[latex]\frac{3^{n}+3^{-n}}{9^{n}+1}=\frac{3^{n}+\frac{1}{3^{n}}}{9^{n}+1}=\frac{3^{2n}+1}{3^{n}(9^{n}+1)}=\frac{9^{n}+1}{3^{n}(9^{n}+1)}=\frac{1}{3^{n}}=3^{-n}} \\\\ \frac{5^{n+1}-5^{n}}{4}=\frac{5^{n}(5-1)}{4}=5^{n}\\\\ \frac{(4^{n}+4^{n-1})^2}{4^{2n-2}}=\frac{(4^{n}(1+\frac{1}{4}))^2}{4^{2n-2}}= \frac{4^{2n}\cdot (\frac{5}{4})^2}{4^{2n}\cdot 4^{-2}}= \frac{5^2}{4^2\cdot 4^{-2}} =5^2=25[/latex]