Question

Let $S_n,S_2,..,$ 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 for which of the following values of n is the are of $S_n$ less than 1 square cm?

• A. 7
• B. 8
• C. 6
• D. 4

Answer 1

The correct option is B 8

Therefore the side of square also be less than 1 cm.
$\therefore I_1=10,I_2=\frac{10}{\sqrt{2}},I_3=\frac{10}{2},I_4=\frac{10}{2\sqrt{2}}{I}_5=\frac{10}{4},I_6=\frac{10}{4\sqrt{2}}{I}_7=\frac{10}{8},I_8=\frac{10}{8\sqrt{2}}$
Hence $I_8=\frac{10}{8\sqrt{2}}<1cm\therefore areaof{S}_8<1c{m}^{2}where{I}_1,I_2,...,I_{8}arethelengthsofsidesofdifferentsquares,fromoutwardstoinwards.$