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 is not the value of $n$ for which the area of $S_n$ less than $1sq.cm$?

• A. $7$
• B. $8$
• C. $9$
• D. $10$

Answer 1

Answer: (B)

$8$

Let $a_n$ denote the length of a side of $S_n$. We are given

Length of a side of $S_n=$ Length of a diagonal of $S_{n+1}$

$\Rightarrow a_n=\sqrt{2}{a}_{n+1}$

$\Rightarrow \frac{a_{n+1}}{a_n}=\frac{1}{\sqrt{2}}$

Thus, $a_1,a_2,a_3,....$ is a G.P. with first term 10 and common ratio $\frac{1}{\sqrt{2}}$.

Therefore,

$a_n=10(\frac{1}{\sqrt{2}}{)}^{n-1}$

Also, Area $\left(S_n\right)=a_n^2<1$

$\Rightarrow {\left[10\right(\frac{1}{\sqrt{2}}{)}^{n-1}]}^2<1$

$\Rightarrow 100<2^{n-1}$

$\Rightarrow$ smallest possible value of n is 8.