Square-triangular number

Statement

Show that $1225$ is at the same time a perfect square and a triangular number: find its square root and the triangular index $n$ with $T_n=1225$.

Solution

Square root: $\sqrt{1225}=\boxed{35},\qquad 35^2=1225.$ Triangular index: solving $\dfrac{n(n+1)}{2}=1225$ gives $n(n+1)=2450=49\cdot 50$, so $n=\boxed{49},\qquad T_{49}=\frac{49\cdot 50}{2}=1225.$