# Sum of Areas of Infinite Number of Squares

**Problem**

The side of a square is 10 m. A second square is formed by joining, in the proper order, the midpoints of the sides of the first square. A third square is formed by joining the midpoints of the second square, and so on. Find the sum of the areas of all the squares if the process will continue indefinitely.

**Solution**

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The areas of squares form an IGP

$a_1 = A_1 = 10^2 ~ \text{m}$

$a_1 = A_1 = 10^2 ~ \text{m}$

$r = \left( \dfrac{5\sqrt{2}}{10} \right)^2 = 1/2$

$S = \dfrac{a_1}{1 - r} = \dfrac{10^2}{1 - 1/2}$

$S = 200 ~ \text{m}^2$ ← *answer*