If $T$ is the $n^{th}$ triangular number, how could you express $T$ in terms of $n$?

What happens if you multiply that expression by $8$ and add $1$?

If you're finding it hard to prove the conjecture, you might like to print out these proof sorter cards, and then cut them out and rearrange them to form a proof. Alternatively, you can use this interactive proof sorter.

For the second conjecture, if you're finding it hard to prove, here is another set of proof sorter cards for you to print out and rearrange, and another interactive proof sorter.