Infinity

In a square with side 18 is inscribed circle, in circle is inscribed next square, again circle and so on to infinity. Calculate the sum of area of all these squares.

Result

S =  648

Solution:

 S1=182 q=12,∣q∣<1 S=S1⋅11−q S=182⋅11−12=648 \ \\ S_1 = 18^2 \ \\ q = \dfrac{ 1}{ 2}, |q|<1 \ \\ S = S_1 \cdot \dfrac{ 1} { 1- q} \ \\ S = 18^2 \cdot \dfrac{ 1} { 1- \dfrac{ 1}{ 2}} = 648

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