Finding the explicit formula for a recursive sequence

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 $$a$$?

• Well, $b_n=3^n$.
– plop
Jun 3 '21 at 16:18
• wow thanks! i did not think about that now its easy, thank you!
– S H
Jun 3 '21 at 16:20

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.