# Expression for finding common elements in two series

I have a series of the form $$S_1=x(x+1)$$ and another series $$S_1/k$$, for any $$k \in \mathbb{N}$$. Now I want to find the values where the elements of two series are equal. For example, let $$k$$ be 3, then the intersection of the two series gives $$2,30,420,5852,81510,1135290$$.

In other words, what is the condition for which a scaled down version of a triangular number is also a triangular number? Can a formula be derived to find these ways without resorting to equating the two series and deriving quadratic roots i.e. using series formulae or triangular numbers etc.?

• Why you need a general term? It's not always possible to have a simple general $n$th term in terms of $n$. – tarit goswami Apr 18 at 14:18