non homogeneous recurrence relation I am trying to solve the non-homogeneous linear recurrence relation:
$$f(n) = 6f(n-1) - 5,\quad  f(0) = 2.$$
How do I go about doing it? This is so different from solving a homogeneous recurrence relation.

Split the solution into a homogeneous solution $f_n^{(H)}$ and inhomogeneous solution $f_n^{(I)}$.  $f_n^{(H)}$ satisfies
$$f_n^{(H)} - 6 f_{n-1}^{(H)} = 0 \implies f_n^{(H)} = A\cdot 6^n$$
$f_n^{(I)}$ is found by a guess; in this case, we can guess that it is a constant, which we find to be $1$ by plugging an unknown constant into the recurrence relation.  Therefore
$$f_n =  A\cdot 6^n + 1$$
We find that $f_0=2 \implies A=1$.  Therefore
$$f_n =  6^n + 1$$
