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Triangles Within Triangles

Age 14 to 16 Challenge Level:


This diagram shows how the first triangular number can be added to 3 copies of the second triangular number to make the fourth triangular number:

A triangle made from three triangular numbers plus one other

That is: $$ T_1 + 3 \times T_2 = T_4 $$ Here is a diagram showing how the second and third triangular numbers can be combined to make the sixth triangular number:

A triangular numebr made from three lots ofone triangular number plus another

$$ T_2 + 3 \times T_3 = T_6 $$ Can you generalise this rule?

Can you find a rule in terms of $ T_n $ and $T_{n+1}$?