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Prove in a sequence exist some non prime number

Source: Mathematics Regional Olympiad of Mexico Southeast 2020 P6

October 23, 2021
number theoryprime numbersSequence

Problem Statement

Prove that for all a,ba, b and x0x_0 positive integers, in the sequence x1,x2,x3,⋯x_1, x_2, x_3, \cdots defined by
xn+1=axn+b,n≥0x_{n+1}=ax_n+b, n\geq 0
Exist an xix_i that is not prime for some i≥1i\geq 1
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