Question

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 $S_1$ to form yet another square $S_3$, and so on. If $A_1$, $A_2$, $A_3$, \dots are the areas and $P_1$, $P_2$, $P_3$, \dots are 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:

• A. $\frac{1 + \sqrt{5}}{a}$
• B. $\frac{2(2 - \sqrt{2})}{a}$
• C. $\frac{2(2 + \sqrt{2})}{a}$
• D. $\frac{2(1 + \sqrt{2})}{a}$

Hint:

In problems involving geometric series, use the sum of infinite geometric series formula to simplify the calculations.

Answer 1

The Correct Option is C

The perimeter of the first square $P_1$ is $4a$. Each subsequent square has its perimeter scaled by a factor of $\sqrt{2}$, so the perimeter of $S_n$ is: \[ P_n = 4a(\sqrt{2})^{n-1} \] The area of the first square $A_1$ is $a^2$, and each subsequent square has its area scaled by a factor of $\frac{1}{2}$, so the area of $S_n$ is: \[ A_n = a^2 \left( \frac{1}{2} \right)^{n-1} \] Now, summing the perimeters and areas: \[ \sum P_n = \frac{4a}{1 - \sqrt{2}} \quad \text{and} \quad \sum A_n = \frac{a^2}{1 - \frac{1}{2}} = 2a^2 \] Thus, the ratio of perimeters to areas is: \[ \frac{P_1 + P_2 + P_3 + \dots}{A_1 + A_2 + A_3 + \dots} = \frac{2(2 + \sqrt{2})}{a} \]