Well done Tom, from Finham Park School, for clear use of notation :

Whenever you add 3 triangles ( as in T₂ ) together with a triangle one size smaller ( as in T₁ ), a new triangle is formed ( T₄ ) , twice the height of the triangle which was used three times ( T₂ ) .

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