Let S1, S2,....be squares such that for each n≥1, the length of a side of Sn equals the length of adiagonal of Sn+1. If the length of a side of S1 is 10 cm, then the smallest value of n for which Area(Sn)

Let $a_{n}$ denote the length of a side of $S_{n} .$ We are given

Length of a side of $S_{n}$ = Length of a diagonal of $S_{n + 1}$

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

Thus, $a_{1}, a_{2}, a_{3}, \dots$ is a G.P. with first term 10 and common ratio $1 / \sqrt{2} .$

Therefore,

$a_{n} = 10 (1 / \sqrt{2})^{n - 1}$

Also, Area $(S_{n}) = a_{n}^{2} < 1$

$\Rightarrow {[10 (1 / \sqrt{2})^{n - 1}]}^{2} < 1 \Rightarrow 100 < 2^{n - 1}$

$\Rightarrow$ smallest possible value of $n$ is 8.

Let $S_{1}, S_{2}, . . . .$ be squares such that for each $n \geq 1,$ 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 the smallest value of $n$ for which Area
( $S_{n}$ ) $< 1$ is