Let $S_1, S_2,....$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 the smallest value of $n$ for which Area

($S_n$) $<1$ is

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 $\left(S_n\right)=a_n^2<1$

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

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