Find the infinite sum of a sequence Define a sequence $a_n$ such that $$a_{n+1}=3a_n+1$$ and $a_1=3$ for $n=1,2,\ldots$. Find the sum $$\sum_{n=1} ^\infty \frac{a_n}{5^n}$$ I am unable to find a general expression for $a_n$. Thanks.

Let $x=\sum_{n=1}^{\infty}\dfrac{a_{n}}{5^n}$,since
$$\dfrac{a_{n+1}}{5^{n+1}}=\dfrac{3}{5}\dfrac{a_{n}}{5^n}+\dfrac{1}{5^{n+1}}$$
so
$$\sum_{n=1}^{\infty}\dfrac{a_{n+1}}{5^{n+1}}=\dfrac{3}{5}\sum_{n=1}^{\infty}\dfrac{a_{n}}{5^n}+\sum_{n=1}^{\infty}\dfrac{1}{5^{n+1}}$$
then we have
$$x-\dfrac{a_{1}}{5}=\dfrac{3}{5}x+\dfrac{\frac{1}{25}}{1-\frac{1}{5}}\Longrightarrow x=\dfrac{3}{8}$$
