Let a, b and n be positive integers and let xn = an + b.
Show that if the sequence x0, x1, x2, ... contains a square of an integer, then it contains infinitely many squares.
Suppose the sequence does contain a square.
Let say xk be the first square, and let it equal r2.
We have r2 = xk = ak + b.
Now, consider that:
(r + ta)2 = r2 + 2rta + t2a2 = ak + b + 2rta + t2a2 = a(k + 2rt + t2a) + b.
So (r + ta)2 = xk + 2rt + t2 is also in the sequence for any positive integer t, and the sequence therefore contains infinitely many squares.