Prove that not all elements of this recurrent sequence are primes This  question  is a part of my assignment on elementary number theory and I am unable to solve it.

Let a, b and $x_0$ be positive integers and $x_n = ax_{n-1} +b $ for n=1, 2,...,... Prove that not all $x_n$ can be primes.

I tried by assuming that let all $x_n$ be primes. Then I can put $x_i$'s recursively to get a, b, $x_0$ but I am not able to find any contradiction based on that.
Can you please help?

If $x_0$ is not prime, then there is nothing to prove.
If $a=1$, then $x_n=x_0 + n b$ and so is a multiple of $x_0$ for $n=x_0$.
If $a\ne1$, then
$$
x_n = x_0 a^n + b\frac{a^n-1}{a-1}
$$
and so $|x_n| \to \infty$ and $n \to \infty$.
If $x_n$ is always prime, then there is an $n$ such that $p=x_n$ does not divide $a$ or $a-1$. We can assume that $n=0$ by ignoring the previous terms. Then $a-1$ is invertible mod $p$. Moreover, if $m$ is a multiple of $p-1$, then $p$ divides $a^m-1$ and and so $p$ divides $x_m$, a contradiction.
