I'm learning to work with polynomials, specifically right now I'm working with progressions like triangular/tetrahedral/pyramid numbers etc.

I know there are different ways to show the nth triangular number ( T(n) ) is

$$
T(n)= \frac{n(n+1)}{2}
$$

Say I wanted to prove this by, first, showing that any formula for the nth triangular must be quadratic. Then I could prove that formula is correct using a Lagrange fit to $(1,1)$, $(2,3)$, $(3,6)$ (which should be unique).

I want to know if there's a way to show that this formula *must be* a polynomial, and that it *must be* 2nd degree, before actually finding that formula. Probably overkill for *this particular* proof, but maybe there's some other problem where that might be useful.