Question

Let $S_1$, $S_2$, $S_3$, ... 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 then 1$c{m}^2$

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

Answer: (B), (C), (D)

The correct options are
B 8
C 9
D 10

The length of a side of $S_n$= length of diagonal of $S_{n+1}$
$\Rightarrow$ The length of a side of $S_n$=$\sqrt{2}$ (length of side of $S_{n+1}$)
$\Rightarrow \frac{lengthofthesideof{S}_{n+1}}{lengthofthesideof{S}_n}=\frac{1}{\sqrt{2}}$; for $n\ge 1$
$\Rightarrow$ Sides of $S_1,S_2,\cdots ,S_n$ form G.P with common ratio $\frac{1}{\sqrt{2}}$ and first term is 10
$\therefore$ Side of $S_n$=10${\left(\frac{1}{\sqrt{2}}\right)}^{n-1}$ $\Rightarrow$ area of $S_n$=${\left(side\right)}^2=\frac{100}{2^{n-1}}$
Now, area of $S_n\le 1$

$\therefore \frac{100}{2^{n-1}}\le 1\Rightarrow 2^{n-1}\ge 100$
$\Rightarrow n-1\ge 7orn\ge 8$
Hence, options (B), (C) and (D) are correct