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 10cm, then for which of the following values of n is the area of $S_n$ less then 1 sq cm

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

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

The correct options are
B 8
C 9
D 10

Given $x_n=x_{n+1}\sqrt{2}$
$\therefore x_1=x_2\sqrt{2},x_2=x_3\sqrt{2},x_n=x_{n+1}\sqrt{2}$
On multiplying $x_1=x_{n+1}(\sqrt{2}{)}^n\Rightarrow x_{n+1}=\frac{x_1}{(\sqrt{2}{)}^n}$
Hence $x_n=\frac{x_1}{(\sqrt{2}{)}^{n-1}}$
Area of $S_n=x_n^2=\frac{x_1^2}{2^{n-1}}<1\Rightarrow 2^{n-1}>x_1^2(x_1=10)$
$\therefore 2^{n-1}>100$
But $2^7>100,2^8>100,$ etc.
$\therefore n-1=7,8,9....\Rightarrow n=8,9,\mathrm{10.....}$