Let A_1, A_2, A_3, …. be squares such that for each n ≥ 1, the length of the side of A_n equals the length of diagonal of A_n +1. If the length of…

Let $\displaystyle \mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3, \ldots$. be squares such that for each $\displaystyle \mathrm{n} \geqslant 1$, the length of the side of $\displaystyle \mathrm{A}_{\mathrm{n}}$ equals the length of diagonal of $\displaystyle \mathrm{A}_{\mathrm{n}+1}$. If the length of $\displaystyle \mathrm{A}_1$ is $\displaystyle 12$ cm, then the smallest value of n for which area of $\displaystyle \mathrm{A}_{\mathrm{n}}$ is less than one, is \_\_\_\_.