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# Triangles Within Squares

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 $T_{n+1}$?

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Age 14 to 16

Challenge Level

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 $T_{n+1}$?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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

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