Recursion squares

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

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