You may also like

problem icon

Triangular Triples

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

problem icon

Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

problem icon

Smith and Jones

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

Triangles Within Squares

Age 14 to 16 Challenge Level:

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 $T_{n+1}$?