Problem: 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 \mathrm{S}_n} is the sum of the areas of the first n circles so inscribed, then, as n grows beyond all bounds, \mathrm{S}_n} approaches:
Answer Choices:
A. 2πm2​
B. 83πm2​
C. 3πm2​
D. 4πm2​
E. 8πm2​
Solution:
The radius of the first circle is 21​m; the side of the second square is 2m​2​; the radius of the second circle is 21​(2m2​​)=22​m​; and so forth. If S is the limiting value of Sn​, then S=π(2m​)2+π(22​m​)2+π(4m​)2+…=πm2(41​+81​+161​+…)=πm2(21​).