<s>[INST] How can I simplify $\frac{3^{n+1} + 3^{n-1}}{3^{n+2} + 3^n}$? This problem got me thinking hard: 
$$\frac{3^{n+1} + 3^{n-1}}{3^{n+2} + 3^n}$$
Considering the fact that the common base is '$3$', I decided to do this:
$$\log_3\left(\frac{(n+1)(n-1)}{(n+2)(n)}\right)$$
$$\log_3\left(\frac{n^2-n+n+1}{n^2+2n}\right)$$
$$\log_3\left(\frac{n^2 + 1}{n^2 + 2n}\right)$$
$$\log_3\left(\frac{1}{2n}\right)$$
I am not 100% sure if have simplified the problem. I need some clarification.
 [/INST] Why are taking logarithm? No need of that. It's just simple Indices.
$\dfrac{3^{n+1} + 3^{n-1}}{3^{n+2} + 3^n}
=\dfrac{3^{n-1}\left(3^2+1\right)}{3^n\left(3^2+1\right)}
=\dfrac{3^{n-1}}{3^n}
=\dfrac{1}{3}$
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