Question

Let $S_1,S_2\dots \dots$ 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 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. 8
• C. 9
• D. 10

Answer 1

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

10

(b), (c), (d)
If a be the side of a square, then $d=a\sqrt{2}$
By given condition $a_n=\sqrt{2}{a}_{n+1}$
or $a_{n+1}=\frac{a_n}{\sqrt{2}}=\frac{a_{n-1}}{(\sqrt{2}{)}^2}=\frac{a_{n-2}}{(\sqrt{2}{)}^3}\dots \dots =\frac{a_1}{(\sqrt{2}{)}^n}$
Replacing n by n-1, we get
$a_n=\frac{a_1}{(\sqrt{2}{)}^{n-1}}=\frac{10}{2^{n-1}/2}$
Area of $S_n<1\Rightarrow a_n^2<1$
$\Rightarrow \frac{100}{2^{n-1}}<1or200<2^{n}or{2}^n>200$
Now $2^7=128<200<2^8=256>200$
$\therefore n=8,9,10\Rightarrow \left(b\right),\left(c\right),\left(d\right)$