General formula of recursive sequence

Question

The recursively defined sequence given by:

a₁=3, a_k+1=4a_k+1 for all k≥1

First 6 terms of the sequence are 3, 13, 53, 213, 853, 3413

I need help in finding the general formula of the sequence.

Answer 1

$a_{k+1}=4a_k+1$

$a_{k}=4a_{k-1}+1$

subtract:

$a_{k+1}-a_k=4a_k-4a_{k-1}$

$a_{k+1}=5a_k-4a_{k-1}$

Characteristic equation is $x^2-5x+4=0.$ Roots are $x=1$ and $4$, so $a_k=A+B4^n$.

Use your knowledge of $a_1$ and $a_2$ to solve for $A$ and $B$.