It starts with a square with side L, to which its diagonal is drawn and that diagonal is the side of a new square. The diagonal is drawn to that new square, which will become the side of another new square. This is continued n times. Find the general formula to calculate the total area and total perimeter of the fractal in the nth iteration

Let's denote the following: $$ L_n $$is the side length of the square in the nth iteration. $$ A_n $$is the total area of the fractal in the nth iteration. $$ P_n $$is the total perimeter of the fractal in the nth iteration. In the first iteration, you start with a square of side length In the first iteration, you start with a square of side length L. For each subsequent iteration, you divide the side length by $$ \sqrt{2}\:\: $$ because the diagonal of square is $$ \sqrt{2} $$ 1. Side Length in the nth iteration: $$ L_n=\frac{L}{\sqrt{2^n}}=\frac{L}{2^{\frac{n}{2}}} $$ 2. Area in the nth iteration: The area of the nth iteration consists of $$ 4^n $$ smaller squares, each with side length $$ L_n $$. So the total area is: $$ A_n=4^n\times\left(L_n\right)^2=4^n\times\left(\frac{L}{2^{\frac{n}{2}}}\right)^2=\frac{L^2\times4^n}{2^n}=L^2\times2^{\frac{n}{2}} $$ 3. Perimeter in the nth iteration: $$ P_n=4^n\times\left(4\times L_n\right)=4^{n+1}\times L_n=4^{n+1}\times\left(\frac{L}{2^{\frac{n}{2}}}\right)=\frac{L\times4^{n+1}}{2^{\frac{n}{2}}} $$$$ =L\times2^{\frac{n}{2}}\times2^{n+1}=2L\times2^{n+1}=2L\times4^{n+1} $$ So, the general formulas for the total area and total perimeter of the fractal in the nth iteration are: Total Area= $$ \left(A_n\right)=L^2\times2^{\frac{n}{2}} $$ Total Perimeter=$$ \left(P_n\right)=2L\times4^{n+1} $$