Question

Let $S_1, S_2, ...$ be the squares such that for each $n \ge 1$, the length of the diagonal of $S_n$ is equal to the length of the side of $S_{n+1}$. If the length of the side of $S_3$ is 4 cm, what is the length of the side of $S_n$?

Accepted Answer

D. Diagonal of $S_n = \sqrt{2} \cdot 	ext{side}(S_n) = 	ext{side}(S_{n+1})$. Thus, the sides form a GP with common ratio $r = \sqrt{2}$. Given $	ext{side}(S_3) = a r^2 = a (\sqrt{2})^2 = 2a = 4 \implies a = 2$. $	ext{side}(S_n) = a r^{n-1} = 2 (\sqrt{2})^{n-1} = 2^1 \cdot 2^{\frac{n-1}{2}} = 2^{\frac{n+1}{2}}$.

Squares Diagonal and Side Length Progression

Explanation

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