If a, a² + 2 and a³ + 10 are in G.P., then find the value(s) of a.

Since a, a² + 2 and a³ + 10 are in G.P.

$\therefore \dfrac{a^2 + 2}{a} = r = \dfrac{a^3 + 10}{a^2 + 2} \\[1em] \Rightarrow \dfrac{a^2 + 2}{a} = \dfrac{a^3 + 10}{a^2 + 2} \\[1em] \Rightarrow (a^2 + 2)^2 = a(a^3 + 10) \\[1em] \Rightarrow a^4 + 4 + 4a^2 = a^4 + 10a \\[1em] \Rightarrow a^4 - a^4 + 4a^2 - 10a + 4 = 0 \\[1em] \Rightarrow 4a^2 - 10a + 4 = 0 \\[1em] \Rightarrow 4a^2 - 8a - 2a + 4 = 0 \\[1em] \Rightarrow 4a(a - 2) - 2(a - 2) = 0 \\[1em] \Rightarrow (4a - 2)(a - 2) = 0 \\[1em] \Rightarrow 4a - 2 = 0 \text{ or } a - 2 = 0 \\[1em] \Rightarrow a = \dfrac{2}{4} \text{ or } a = 2. \\[1em] \Rightarrow a = \dfrac{1}{2} \text{ or } a = 2.$

Hence, the required value(s) of a are $\dfrac{1}{2}$ and 2.