Solving recurrence $X_n = 4X_{n-1}+5$ $X_n=4X_{n-1}+5$

How come the solution of this recurrence is this? 

$X_n=\frac834^n+\frac53$

I also have that $X_0=1$.

I am using telescoping method and I am trying to solve it like this:

$X_n= 5 + 4X_{n-1}$

$X_n= 5 + 4(5+4X_{n-2})$

$X_n= 5 + 4\times5 + 4\times4\times X_{n-2}$

But this leads to me getting $5\times4^{n-1}\times4^n$.

Can some please explain this to me?
If you calculate the first few terms explicitly, you will find that the $n$th term is the sum of an exponential and a geometric series. For example,

$$
X_3 = 4^3 + 5(4^2+4+1).
$$
So in general,
$$
X_n = 4^n + 5\sum_{k=0}^{n-1}4^k = 4^n +5\frac{1-4^n}{1-4},
$$

which should simplify to the answer you gave.
This is a difference equation and it can be solved using Z-Transform. Take Z-transform of both sides of the equation and then use the initial condition. Ultimately, you will get Z-transform of X and then take its inverse z-transform to get the solution.