Multiple Choice

Let $$S_1, S_2, ... $$ be squares such that for each $$n\geq 1$$, 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$$8$$
Correct Answer
C$$9$$
Correct Answer
D$$10$$
Correct Answer

Solution

Let $$a_n$$ denotes the side of the square $$S_n$$ then
$$a_n=\sqrt 2a_{n+1}$$
$$\Rightarrow \dfrac {a_{n+1}}{a_n}=\dfrac {1}{\sqrt 2}$$
$$\Rightarrow a_n=a_1(\dfrac {1}{\sqrt 2})^{n-1}$$ (G.P. formula)$$=10(\dfrac {1}{\sqrt 2})^{n-1}$$
Now, we must have $$a_n^2 < 1$$
$$\Rightarrow 100(\dfrac {1}{\sqrt 2})^{2n-1} < 1$$
$$\Rightarrow 2^n > 200 \Rightarrow n > 7$$

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