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 = 20 cm.

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

Final Answer:

s₁ =  273.14 cm
s₂ =  800 cm²

Step-by-step explanation:

$$\begin{gathered} a_1=20\text{cm} \\ {a}_2=\sqrt{(a_1/2{)}^2+(a_1/2{)}^2}=\sqrt{(20/2{)}^2+(20/2{)}^2}=10\sqrt{2}\text{cm}\doteq 14.1421\text{cm} \\ \dots \\ {p}_1=4\cdot a_1=4\cdot 20=80\text{cm} \\ {p}_2=4\cdot a_2=4\cdot 14.1421=40\sqrt{2}\text{cm}\doteq 56.5685\text{cm} \\ \dots \\ {q}_1=p_2/p_1=56.5685/80\doteq 0.7071 \\ {s}_1=p_1\cdot \frac{1}{1-q_1}=80\cdot \frac{1}{1-0.7071}=273.14\text{cm} \end{gathered}$$

$$\begin{gathered} S_1=a_1^2=20^2=400{\text{cm}}^2 \\ {S}_2=a_2^2=14.1421^2=200{\text{cm}}^2 \\ \dots \\ {q}_2=S_2/S_1=200/400=\frac{1}{2}=0.5 \\ {s}_2=S_1\cdot \frac{1}{1-q_2}=400\cdot \frac{1}{1-0.5}=800{\text{cm}}^2 \end{gathered}$$

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