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'Triangles Within Triangles' printed from https://nrich.maths.org/
Well done Tom, from Finham Park School, for
clear use of notation :
Whenever you add 3 triangles ( as in $T_2$ ) together with a
triangle one size smaller ( as in $T_1$ ), a new triangle is formed
( $T_4$ ) , twice the height of the triangle which was used three
times ( $T_2$ ) .
The smaller triangle can be called $T_n$ , while the 3 triangles
one size up can be called $T_{n+1}$.
One of the $T_{n+1 }$ joins with the $T_n $ to form a square of
side length $n+1$ .
The two remaining $T_{n+1}$ fit to that square producing a large
triangle that has a height twice that of $T_{n+1}$ ,
So the sum of all four triangles is the triangle $T_{2(n+1)}$
So $T_n $ + $3T_{n+1} =T_{2(n+1)}$ or, if you prefer,
$T_{2n+2}$