The diagram above shows that:

8 \times T_{2} + 1 = 25 = 5^{2}

Use a similar method to help you verify that:

8 \times T_{3} + 1 = 49 = 7^{2}

Can you generalise this result?

Can you find a rule in terms of $T_{n}$ and a related square number?

Can you find a similar rule involving square numbers for $T_{n}, T_{n + 2}$ and several copies of $T_{n + 1}$ ?