If the first term of a G.P. is 5 and the sum of first three terms is $\dfrac{31}{5},$ find the common ratio.

Given, a = 5 and S₃ = $\dfrac{31}{5}$.

Using formula $S_n = \dfrac{a(r^n - 1)}{r - 1}$ we get,

$\Rightarrow S_3 = \dfrac{5(r^3 - 1)}{r - 1} \\[1em] \Rightarrow \dfrac{31}{5} = \dfrac{5(r^3 - 1)}{r - 1} \\[1em] \Rightarrow 31(r - 1) = 25(r^3 - 1) \\[1em] \Rightarrow 25(r - 1)(r^2 + r + 1) = 31(r - 1) \\[1em]$

Dividing by (r - 1) on both sides we get,

$\Rightarrow 25(r^2 + r + 1) = 31 \\[1em] \Rightarrow 25r^2 + 25r + 25 = 31 \\[1em] \Rightarrow 25r^2 + 25r + 25 - 31 = 0 \\[1em] \Rightarrow 25r^2 + 25r - 6 = 0 \\[1em] \Rightarrow 25r^2 + 30r - 5r - 6 = 0 \\[1em] \Rightarrow 5r(5r + 6) - 1(5r + 6) = 0 \\[1em] \Rightarrow (5r - 1)(5r + 6) = 0 \\[1em] \Rightarrow 5r - 1 = 0 \text{ or } 5r + 6 = 0 \\[1em] \Rightarrow 5r = 1 \text{ or } 5r = -6 \\[1em] \Rightarrow r = \dfrac{1}{5} \text{ or } r = -\dfrac{6}{5}.$

Hence, the common ratio of the G.P. is $\dfrac{1}{5} \text{ or } -\dfrac{6}{5}.$