Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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if $x_{n+1}=ax_{n}+b$ then How find this $a,b,c$?

Question: Let there be a sequence $x_{n}$ such that $$x_{n+1}=ax_{n}+b,,x_{1}=c$$ where $a,b$ and $c$ are positive integers. Suppose, too, that for $n,m\in \mathbb{Z^{+}}$ we have that $$ ...