Question

Let $S_1$, $S_2$,....... be squares such that for each n≥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 area of $S_n$ greater then 1sq cm

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

The correct option is A

7

(b, c, d) Given $x_n$ = $x_{n+1}$ $\sqrt{2}$

∴ $x_1$ = $x_2$$\sqrt{2}$, $x_2$ = $x_3$$\sqrt{2}$, $x_n$ = $x_{n+1}$$\sqrt{2}$

On multiplying $x_1$ = $x_{n+1}$ ${\left(\sqrt{2}\right)}^n$ ⇒ $x_{n+1}$ = $\frac{x_1}{(\sqrt{2}{)}^n}$

Hence $x_n$ = $\frac{x_1}{(\sqrt{2}{)}^{n-1}}$

Area of $S_n$ = $x_n^2$ = $\frac{x_n^2}{2^{n-1}}$ < 1 ⇒ $2^{n-1}$ > $x_1^2$ ($x_1$ = 10)

∴ $2^{n-1}$ > 100

But $2^7$ > 100, $2^8$>100, etc.

∴ n - 1 = 7, 8, 9....... ⇒ n = 8, 9, 10.........