Question

Sol. [Key Concept: Three G.P. terms form a triangle ⇒ triangle inequality bounds the common ratio.]

Let $a,ar,{ar}^{2}$ be the sides $r > 0,a > 0$

$$\begin{matrix} a(1 + r) > {ar}^{2}\ \Rightarrow r^{2} - r - 1 < 0 & & \Rightarrow \ r \in \left( 0,\frac{1 + \sqrt{5}}{2} \right) \\ a\left( 1 + r^{2} \right) > ar\ \Rightarrow r > 0 & & \Rightarrow \ \Rightarrow \ r \in \left( \frac{- 1 + \sqrt{5}}{2},\infty \right) \\ a\left( r^{2} + r \right) > a\ \Rightarrow r^{2} + r - 1 > 0 & \Rightarrow \ r \in \left( \frac{\sqrt{5} - 1}{2},\frac{\sqrt{5} + 1}{2} \right) & \end{matrix}$$

✘ (a) $\lbrack r\rbrack \neq 2$

✔ (b) $r \in (0.618,1) \Rightarrow \lbrack r\rbrack = 0$

✔ (c) $r \in \lbrack 1,1.618) \Rightarrow \lbrack r\rbrack = 1$

✘ (d) $\lbrack r\rbrack \neq 3$