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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: $$ 8 \times T_2 + 1 = 25 = 5^2$$

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 $T_{n+1}$?