Infinitely many squares in a sequence.

Question

In the sequence of integers $a, a+d, a + 2d, a + 3d, ... $

$a,d> 0$

How to prove that if one of the numbers is a square, than there are infinitely many squares in the sequence ?

Answer 1

If $b^2$ is any square and any $d > 0$ is given, then for any $n > 0$, $$b^2 + (dn^2+2nb)d = (b+dn)^2$$ is also a square.

Answer 2

Hint: WLOG let $a$ be a square. Then can you show $(\sqrt{a}+d)^2$ appears in the sequence?