Image with six copies of the second triangular number and one of hte first triangular number added to make a square

The diagram above shows that:
8 ×T2 + 1 = 25 = 52

Use a similar method to help you verify that:
8 ×T3 + 1 = 49 = 72
Can you generalise this result?

Can you find a rule in terms of Tn and a related square number?

Can you find a similar rule involving square numbers for Tn, Tn+2 and several copies of Tn+1?