Question

Let $S_1, S_2, S_3, \ldots \ldots \ldots$. be squares such that for each $n \geq 1$, 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 $1 \,cm ^2$ ?

• A. 7
• B. 8
• C. 6
• D. 5

Answer 1

Answer: (B)

If $a$ be the side of a square then $d=a \sqrt{2}$
by given condition $\rightarrow a_n=\sqrt{2} a_{n+1}$
or $a_{n+1}=\frac{a_n}{\sqrt{2}}=\frac{a_{n-1}}{(\sqrt{2})^2}=\frac{a_{n-2}}{(\sqrt{2})^3}=\ldots . .=\frac{a_1}{(\sqrt{2})^n}$
Replacing $n$ by $n-1$, we get
$a_n=\frac{a_1}{(\sqrt{2})^{n-1}}=\frac{10}{2^{\frac{(n-1)}{2}}}$
Area of $S_n<1 \Rightarrow a_n^2<1$
$\Rightarrow \quad \frac{100}{2^{n-1}}<1$ or $200<2^n$ or $2^n>200$
Now $2^7=128<200,2^8=256>200$
$\therefore n =8,9,10$