First slide

Geometric progression

Question

The value of common ratio for which three successive terms of a GP are the sides of a triangle is

Moderate

A

$r > 0$
B

$r \in (0, 1)$
C

$r \in (\frac{\sqrt{5} - 1}{2}, \frac{\sqrt{5} + 1}{2})$
D

None of these

Solution

Let r be the common ratio. If r $\geq$ 1, then ar² is the greatest term.
$\begin{array}{lrl} ∴ {ar}^{2} < a + ar \\ \Rightarrow r^{2} - r - 1 < 0 \\ \Rightarrow r = \frac{1 \pm \sqrt{1 + 4}}{2} \\ \Rightarrow r = \frac{1 \pm \sqrt{5}}{2} \\ \Rightarrow r \in (\frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2}) \\ As r \geq 1, 1 \leq r < \frac{1}{2} (\sqrt{5} + 1) \\ In case, 0 < r < 1 \\ \Rightarrow a < ar + {ar}^{2} \\ \Rightarrow r^{2} + r - 1 > 0 \\ o r = \frac{- 1 \pm \sqrt{5}}{2} \Rightarrow r < - \frac{1}{2} (\sqrt{5} + 1) \end{array}$
$\begin{array}{ll} As 0 < r < 1 \Rightarrow \frac{\sqrt{5} - 1}{2} < r < 1 \\ ∴ r \in (\frac{\sqrt{5} - 1}{2}, \frac{\sqrt{5} + 1}{2}) \end{array}$

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