Copyright © University of Cambridge. All rights reserved.

'Triangles Within Squares' printed from https://nrich.maths.org/

Show menu


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 $T_{n+1}$?