Finding the explicit formula for a recursive sequence

Question

There is a recursive sequence defined by: $a_1=1$ and $a_{n+1} = a_n/(4\cdot a_n + 3)$. I've also been given this sequence: $b_n = (2\cdot a_n + 1)/a_n$ and I have found that $b_{n+1} = 3 \cdot b_n$. How can I find the explicit formula of $a$?

Answer 1

In my answer to this question, I detailed the steps for solving a first-order rational difference equation such as $${ a_{n+1} = \frac{m\,a_n + x}{a_n + y} }$$

For your case $m=\frac 14$, $x=0$, $y=\frac 34$ makes the problem quite simple. Just follow the steps.