The following statement is to be proven or disproven:
Let denote the triangular numbers.
If is also a triangular number for all integer values then is a perfect square, and is a triangular number.
Can be a perfect square of an even number?
If is a triangular number, we may generalize the formula as where are natural numbers.
Substitute the above expression for to the LHS and expand the RHS. Multiply both sides by 2 to get
By equating like terms on both sides, we discover that A is a perfect square and B is a triangular number:
Since , we find that , an odd number.
Therefore, is a perfect square of an odd number, and is a triangular number.
* As an exercise, prove that .*
Check out my other notes at Proof, Disproof, and Derivation