<s>[INST] Problems with Recurrence Relations as a form of Counting I have been having trouble trying to understand how to do the following problem

Solve by unfolding: $a_0=3$, and for $n\geq1$, $a_n=5a_{n-1}+3$. Hint: This will involve the geometric sum formula.


This is my work so far:
$$a_n=5a_{n-1}+3$$
$$a_n=5(5a_{n-2}+3)+3$$
$$a_n=(5(5(5a_{n-3}+3)+3)+3)$$
$$a_n=5^{n}*a_0+5^{n-1}*3+5^{n-2}*3+...+5*3+3$$
I am not sure if this is right, or how to really do this problem. Help, and hints would be much appreciated. 
 [/INST] \begin{align}
a_n
&= 5a_{n-1}+3\\
&= 5(5a_{n-2}+3)+3\\
&= 5^2a_{n-2}+3(5^1+5^0)\\
&= 5^2(5a_{n-3}+3)+3(5^1+5^0)\\
&= 5^3a_{n-3}+3(5^2+5^1+5^0)\\
&= \dots\\
&= 5^na_0+3(5^{n-1}+\dots+5^0)\\
&= 5^n(3)+3\cdot\frac{5^n-1}{5-1}\\
&= 3\cdot5^n+\frac34(5^n-1)\\
&= \frac{15}4\cdot5^n-\frac34\\
&= \frac34(5^{n+1}-1)
\end{align}
</s>