1967 AMC 12 Question 20 - thichhoctoan.com

Question 20

A circle is inscribed in a square of side $m,$ then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches:

$\dfrac{\pi m^2}{2}$
$\dfrac{3\pi m^2}{8}$
$\dfrac{\pi m^2}{3}$
$\dfrac{\pi m^2}{4}$
$\dfrac{\pi m^2}{8}$