The fifth term of a G.P. is 81 and its second term is 24. Find the geometric progression.

Let first term of the G.P. be a and it's common ratio be r.

Given,

⇒ a₅ = 81

⇒ ar⁴ = 81 ……..(i)

Also,

⇒ a₂ = 24

⇒ ar = 24 ……..(ii)

Dividing (i) by (ii) we get,

$\Rightarrow \dfrac{ar^4}{ar} = \dfrac{81}{24} \\[1em] \Rightarrow r^3 = \dfrac{27}{8} \\[1em] \Rightarrow r^3 = \Big(\dfrac{3}{2}\Big)^3 \\[1em] \Rightarrow r = \dfrac{3}{2}.$

Substituting value of r in (ii) we get,

$\Rightarrow a \times \dfrac{3}{2} = 24 \\[1em] \Rightarrow a = \dfrac{2}{3} \times 24 \\[1em] \Rightarrow a = 16.$

⇒ a₃ = ar²

= $16 \times \Big(\dfrac{3}{2}\Big)^2$

= $16 \times \dfrac{9}{4}$

= 4 × 9

= 36.

⇒ a₄ = ar³

= $16 \times \Big(\dfrac{3}{2}\Big)^3$

= $16 \times \dfrac{27}{8}$

= 2 × 27

= 54.

G.P. = 16, 24, 36, 54, 81, ………..

Hence, G.P. = 16, 24, 36, 54, 81, ………..