Recognizing a triangular number

Statement

A triangular number has the form $T_n=\dfrac{n(n+1)}{2}$. Decide whether $435$ is triangular and, if so, find the index $n$.

Solution

An integer $t$ is triangular iff $8t+1$ is an (odd) perfect square, since $8T_n+1=(2n+1)^2$. Here $8\cdot 435+1=3481=59^2,$ so $435$ is triangular. From $2n+1=59$ we get $n=\boxed{29},\qquad T_{29}=\frac{29\cdot 30}{2}=435.$