The sum of first three terms of a G.P. is $\dfrac{39}{10}$ and their product is 1. Find the common ratio and the terms.

Given, S₃ = $\dfrac{39}{10}$ and a₁ × a₂ × a₃ = 1.

Let the three numbers that are in G.P. be $\dfrac{a}{r}, a, ar.$

$\therefore \dfrac{a}{r} \times a \times ar = 1 \\[1em] \Rightarrow a^3 = 1 \\[1em] \Rightarrow a = 1. \\[1em] S_3 = \dfrac{39}{10} \\[1em] \Rightarrow \dfrac{a}{r} + a + ar = \dfrac{39}{10} \\[1em] \Rightarrow a\Big(\dfrac{1}{r} + 1 + r\Big) = \dfrac{39}{10} \\[1em] \Rightarrow 1\Big(\dfrac{1 + r + r^2}{r}\Big) = \dfrac{39}{10} \\[1em] \Rightarrow 10(1 + r + r^2) = 39r \\[1em] \Rightarrow 10 + 10r + 10r^2 = 39r \\[1em] \Rightarrow 10r^2 + 10r - 39r + 10 = 0 \\[1em] \Rightarrow 10r^2 - 29r + 10 = 0 \\[1em] \Rightarrow 10r^2 - 25r - 4r + 10 = 0 \\[1em] \Rightarrow 5r(2r - 5) - 2(2r - 5) = 0 \\[1em] \Rightarrow (5r - 2)(2r - 5) = 0 \\[1em] \Rightarrow 5r - 2 = 0 \text{ or } 2r - 5 = 0 \\[1em] \Rightarrow r = \dfrac{2}{5} \text{ or } r = \dfrac{5}{2} \\[1em] \text{First taking r } = \dfrac{2}{5} \\[1em] \therefore \text{Terms are : } \dfrac{a}{r} = \dfrac{1}{\dfrac{2}{5}} = \dfrac{5}{2}, a = 1, ar = 1 \times \dfrac{2}{5} = \dfrac{2}{5}. \\[1em] \text{Now, taking r } = \dfrac{5}{2} \\[1em] \therefore \text{Terms are : } \dfrac{a}{r} = \dfrac{1}{\dfrac{5}{2}} = \dfrac{2}{5}, a = 1, ar = 1 \times \dfrac{5}{2} = \dfrac{5}{2}. \\[1em]$

Hence, common ratio is $\dfrac{2}{5}$ or $\dfrac{5}{2}$ and terms are $\dfrac{5}{2}, 1, \dfrac{2}{5}$ or $\dfrac{2}{5}, 1, \dfrac{5}{2}.$