First slide

Equation of line in Straight lines

Question

Let S₁, S₂, ... be squares such that for each n ≥ 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 S₁ is 10 cm, then for which of the following values of n is the area of S_n less than 1 square cm?

Moderate

A

7
B

8
C

9
D

10

Solution

Let a be the side of the square, then diagonal $d = a \sqrt{2}$
Given, $a_{n} = \sqrt{2} a_{n + 1}$
$\Rightarrow a_{n + 1} = \frac{a_{n}}{\sqrt{2}} = \frac{a_{n - 1}}{{(\sqrt{2})}^{2}} = \frac{a_{n - 2}}{{(\sqrt{2})}^{3}} = . . . . = \frac{a_{1}}{{(\sqrt{2})}^{n}}$
$∴ a_{n + 1} = \frac{a_{1}}{{(\sqrt{2})}^{n}} \Rightarrow a_{n} = \frac{a_{1}}{{(\sqrt{2})}^{n - 1}} = \frac{10}{2^{\frac{n - 1}{2}}}$
Now, area of $s_{n} < 1 \Rightarrow a_{n}^{2} < 1 \Rightarrow \frac{100}{2^{n - 1}} < 1$
$\Rightarrow 2^{n} > 200 > 2^{7} \Rightarrow n > 7$

∴ n = 8, 9, 10.

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