Find $n$ so that $9^{n+2} = 240 + 9^n$.

$9^{n+2} = 240 + 9^n\\[1em] \Rightarrow 9^{n+2} - 9^{n} = 240\\[1em] \Rightarrow 9^{n}[9^2 - 1] = 240\\[1em] \Rightarrow 9^n[81 - 1] = 240\\[1em] \Rightarrow 9^n[80] = 240\\[1em] \Rightarrow 9^n = \dfrac{240}{80}\\[1em] \Rightarrow 9^n = 3\\[1em] \Rightarrow 3^{2n} = 3^1 \\[1em] \Rightarrow 2n = 1 \\[1em] \Rightarrow n = \dfrac{1}{2}$

If $9^{n+2} = 240 + 9^n$, then $n = \dfrac{1}{2}$.