Question

We have seen how we can draw a series of right triangles as in the picture.
How do we wire in algebra, the difference in perimeter of the $n^{th}$ triangle and that of the triangle just be fore it?

Answer 1

Perimeter of $n^{th}\Delta$

$=\sqrt{n}+1+\sqrt{n+1}$

Perimeter of $(n-1{)}^{th}\Delta$

$=\sqrt{n-1}+1+\sqrt{n^2-2n+1+1}$

$(n{)}^{th}-(n-1{)}^{th}=\sqrt{n}+\sqrt{n+1}-\sqrt{n-1}-\sqrt{n^2-2n+2}$

Length of tenth $\Delta$

$=\sqrt{10},1,\sqrt{11}$