Recursion squares

In the square, ABCD has inscribed a square so that its vertices lie at the centers of the sides of the square ABCD. The procedure of inscribing the square is repeated this way. The side length of the square ABCD is a = 22 cm.

Calculate:
a) the sum of perimeters of all squares
b) the sum of the area of all squares

Correct result:

Σ p =  300.45 cm
Σ S =  968 cm²

Solution:

$$\begin{gathered} p_1=4\cdot 22 \\ {q}_p={\displaystyle \frac{1}{\sqrt{2}}} \\ \mathrm{\Sigma }p=4\cdot 22\cdot {\displaystyle \frac{1}{1-{\displaystyle \frac{1}{\sqrt{2}}}}}=300.45\text{ cm} \end{gathered}$$

$$\begin{gathered} {S}_1=22^2 \\ {q}_S={\displaystyle \frac{1}{2}} \\ \mathrm{\Sigma }S=22^2\cdot {\displaystyle \frac{1}{1-{\displaystyle \frac{1}{2}}}}=968{\text{cm}}^2 \end{gathered}$$

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