Question

Let $S_1,S_2,...$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals to 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 value(s) of n is the area of $S_n$ less than 1 sq. cm?

• A. $7$
• B. $8$
• C. $9$
• D. $10$

Answer 1

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

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

Let $a_n$ denotes the side of the square $S_n$ then

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

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

$\Rightarrow a_n=a_1(\frac{1}{\sqrt{2}}{)}^{n-1}$ (G.P. formula)$=10(\frac{1}{\sqrt{2}}{)}^{n-1}$

Now, we must have $a_n^2<1$

$\Rightarrow 100(\frac{1}{\sqrt{2}}{)}^{2n-1}<1$

$\Rightarrow 2^n>200\Rightarrow n>7$