Infinity

In a square with side 19 is inscribed circle, the circle is inscribed next square, again circle, and so on to infinity. Calculate the sum of the area of all these squares.

Correct answer:

S =  722

Step-by-step explanation:

$$\begin{gathered} a_1=19 \\ {S}_1=a_1^2=19^2=361 \\ {d}_1=a_1=19 \\ {d}_1=\sqrt{2}{a}_2 \\ {a}_2=d_1/\sqrt{2}=19/\sqrt{2}\doteq 13.435 \\ {S}_2=a_2^2=13.435^2=\frac{361}{2}=180\frac{1}{2}=180.5 \\ q=S_2/S_1=180.5/361=\frac{1}{2}=0.5 \\ S=S_1\cdot \frac{1}{1-q}=361\cdot \frac{1}{1-0.5}=722 \\ \text{ Verifying Solution: } \\ {d}_2=a_2=13.435\doteq 13.435 \\ {d}_2=\sqrt{2}{a}_3 \\ {a}_3=d_2/\sqrt{2}=13.435/\sqrt{2}=\frac{19}{2}=9\frac{1}{2}=9.5 \\ {S}_3=a_3^2=9.5^2=\frac{361}{4}=90\frac{1}{4}=90.25 \\ {q}_2=S_3/S_2=90.25/180.5=\frac{1}{2}=0.5 \end{gathered}$$

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