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