Let $S_1$ be a square of side a. Another square $S_2$ is formed by joining the mid-points of the sides of $S_1$. The same process is applied to $S_2$ to form yet another square $S_3$, and so on. If $A_1, A_2, A_3, \dots$ be the areas and $P_1, P_2, P_3, \dots.$ be the perimeters of $S_1, S_2, S_3, \dots.$, respectively, then the ratio $\frac{P_1 + P_2 + P_3 + \dots}{A_1+ A_2+ A_3 + \dots}$ equals
- $\frac{2(1+\sqrt{2})}{a}$
- $\frac{2(2-\sqrt{2})}{a}$
- $\frac{2(2+\sqrt{2})}{a}$
- $\frac{2(1+2\sqrt{2})}{a}$