Question

Let $S_1, S_2,...$ be squares such that for each $n \ge 1$ the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is 10 cm, then for which of the following values of n is the area of $S_n$ less than 1 sq cm?

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

Answer: (D)

10

Answer 2

Let $a_n$ denotes the length of side of the square $S_n$.
We are given, $a_n$ = length of diagonal of $S_{n+1}.$
\Rightarrow \hspace20mm a_n = \sqrt 2\, a_{n+1}
\Rightarrow \hspace20mm a_{n+1} = \frac{a_n}{\sqrt 2}
This shows that $a_1 ,a_2, a_3,...$ form a GP with common ratio $1/ \sqrt 2.$
Therefore, $a_n = a_1 \bigg(\frac{1}{\sqrt 2}\bigg)^{n-1}$
\Rightarrow \hspace10mm a_n = 10 \bigg(\frac{1}{\sqrt 2}\bigg)^{n-1} \hspace10mm [\because a_1 = 10, given]
\Rightarrow \hspace10mm a^2_n = 100 \bigg(\frac{1}{\sqrt 2}\bigg)^{2(n-1)}
\Rightarrow \hspace10mm \frac{100}{2^{n-1}} \le 1 \hspace20mm [\because a^2_n \le 1, given]
\Rightarrow \hspace10mm 100 \le 2^{n-1}
This is possible for $n \ge 8$.
Hence, (b), (c), (d) are the correct answers.