How to find the closed form of this recurrence relation?

Question

The relation is :

$x(n) = x(n-1)*c + z $,

where $c$ and $z$ are constants , $x[1]$ and $x[2]$ are known .

$c = 2*p - 1$ ;

$z = 1 - p $

$p$ is a constant .

Answer 1

A constant solution is $x(n)=\frac z{1-c}$, so we expect the substitution $x(n)=\frac z{1-c}+y(n)$ to simplify things. Indeed, this yields $y(n)=cy(n-1)$, so $y(n)=c^ny(0)$, so $x(n)=\frac z{1-c}+c^n\left(x(0)-\frac z{1-c}\right)$.