The product of 3^rd term and 8^th terms of a G.P. is 243. If its 4^th term is 3, find its 7^th term.

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

Given,

⇒ a₃.a₈ = 243

⇒ ar².ar⁷ = 243

⇒ a²r⁹ = 243 ……….(i)

Also,

⇒ a₄ = 3

⇒ ar³ = 3

⇒ a = $\dfrac{3}{r^3}$ ……..(ii)

Substituting value of a from (ii) in (i) we get,

$\Rightarrow \Big(\dfrac{3}{r^3}\Big)^2 \times r^9 = 243 \\[1em] \Rightarrow \dfrac{9}{r^6} \times r^9 = 243 \\[1em] \Rightarrow 9r^3 = 243 \\[1em] \Rightarrow r^3 = 27 \\[1em] \Rightarrow r = \sqrt[3]{27} \\[1em] \Rightarrow r = 3.$

Substituting value of r in (ii),

$\Rightarrow a = \dfrac{3}{r^3} \\[1em] = \dfrac{3}{3^3} \\[1em] = \dfrac{1}{3^2} \\[1em] = \dfrac{1}{9}.$

a₇ = ar⁶

= $\dfrac{1}{9} \times 3^6$

= $\dfrac{1}{9} \times 729$

= 81.

Hence, 7^th term = 81.