Question

Let $S_1,S_2,S_3,\dots \dots \dots$. 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 than $1c{m}^2$ ?

• 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 $\to a_n=\sqrt{2}{a}_{n+1}$
or $a_{n+1}=\frac{a_n}{\sqrt{2}}=\frac{a_{n-1}}{(\sqrt{2}{)}^2}=\frac{a_{n-2}}{(\sqrt{2}{)}^3}=\dots ..=\frac{a_1}{(\sqrt{2}{)}^n}$
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 $S_n<1\Rightarrow a_n^2<1$
$\Rightarrow \frac{100}{2^{n-1}}<1$ or $200<2^n$ or $2^n>200$
Now $2^7=128<200,2^8=256>200$
$\therefore n=8,9,10$