Question

Let $S_1,S_2,S_3,\dots ,S_n$ 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 than $1$ sq. cm?

• A. $7$
• B. $9$
• C. $8$
• D. $10$

Answer 1

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

The correct options are
B $9$
C $8$
D $10$

If $a$ be the side of the square, then $d=a\sqrt{2}$.
By given condition, $a_n=\sqrt{2}{a}_{n+1}$
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
$a_n=\frac{a_1}{(\sqrt{2}{)}^{n-1}}=\frac{10}{2^{(n-1)/2}}$
Area of $S_n<1\Rightarrow a_n^2<1$
$\Rightarrow \frac{100}{2^{n-1}}<1$
$\Rightarrow 2^n>200$
$\therefore n=8,9,10$