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.

HINT:
Let $a_m=b_m+c\implies b_{n+1}+c=3(b_n+c)+1\iff b_{n+1}=b_n+2c+1$ 
Set $2c+1=0$ to get
$$a_{n+1}+\dfrac12=3^1\left[a_n+\dfrac12\right]=\cdots=3^r\left[a_{n-r+1}+\dfrac12\right]=3^n\left[a_1+\dfrac12\right]$$
