# IFF - Interactive Proof Sorter

##### Age 14 to 18 Challenge Level:
something something sorted not sorted

Then $T=\frac12n(n+1)$ for some whole number $n$

Expanding, $8T+1=4n^2+4n+1$

Let $T$ be a triangular number

Therefore, if $T$ is triangular, $8T+1$ is square

We wish to prove that if $T$ is a triangular number then $8T+1$ is a square number.

Simplifying, $8T+1=4n(n+1)+1$

Factorising the right hand side, $8T+1=(2n+1)^2$

Therefore $8T+1 = 8(\frac12n(n+1))+1$