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Let $S_{1}, S_{2}, .......$ be the squares such that for each $n \geq 1$ the length of the side S_n is equal to length of the diagonal of $S_{n + 1}$ .If the length of the side of $S_{1}$ is 10 cm ,then the least value of n for which area of $S_{n} < 1$ sq.cm given by

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Detailed Solution

Side of the square $s_{1} = 10 c m$
Diagonal of the square $s_{2} = 10 c m$
side of the square $s_{2} = \frac{10}{\sqrt{2}} c m$ .
similarly side of the square $s_{3} = \frac{10}{{(\sqrt{2})}^{2}} c m$
side of the square $s_{n} = \frac{10}{{(\sqrt{2})}^{n - 1}}$
But area of $s_{n} < 1 \Rightarrow$ ${(\frac{10}{{(\sqrt{2})}^{n - 1}})}^{2} < 1$ ;
$\Rightarrow$ 100 < $2^{n - 1}$

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