Let's consider an arithmetical or geometrical progression with all elements natural numbers, which starts with a perfect square. Prove that the progression includes an infinity of perfect squares!

It is direct for a Geometric Progression. If the 1st is perfect square
then 3rd,5th all the odd terms will be perfect squares.

Let for Arithmetic Progression starts with a² and common
difference is d.
Consider (a+nd)², n=0,1,2,..
all these terms will be in Arithmetic Progression
starting with a² and there are infinite such terms
hence proved.

Posted by Praneeth on 2007-12-26 01:10:42