In a Geometric Progression (G.P.) the first term is 24 and the fifth term is 8. Find the ninth term of the G.P.

Let first term of G.P. be a and common ratio be r.

Given,

First term (a) = 24

Fifth term (ar⁴) = 8

$$\begin{aligned}\Rightarrow \frac{ar^4}{a} = \frac{8}{24} \\[1em] \Rightarrow r^4 = \frac{1}{3} \Rightarrow (r^4)^2 = \Big(\frac{1}{3}\Big)^2 \\[1em] \Rightarrow r^8 = \frac{1}{9}.\end{aligned}$$

By formula,

Ninth term of G.P. (a₉) = ar⁸

= $24 \times \frac{1}{9} = \frac{8}{3}$.

Hence, ninth term of G.P. = $\frac{8}{3}$.