Find the G.P. whose 5th and 8th terms are 80 and 640 respectively.

Let a be the first term and r be the common ratio.

We know that,

nth term of a G.P. is given by,

T_n = ar^{n - 1}

Given,

5th term = 80

⇒ ar^{5 - 1} = 80

⇒ ar⁴ = 80 ….(1)

Given,

8th term = 640

⇒ ar^{8 - 1} = 640

⇒ ar⁷ = 640 …….(2)

Dividing Equation (2) by Equation (1) :

$\Rightarrow \dfrac{ar^7}{ar^4} = \dfrac{640}{80} \\[1em] \Rightarrow r^{7 - 4} = 8 \\[1em] \Rightarrow r^{3} = 8 \\[1em] \Rightarrow r = \sqrt[3]8 \\[1em] \Rightarrow r = 2.$

Substituting r = 2 in Equation (1), we get :

⇒ a(2)⁴ = 80

⇒ 16a = 80

⇒ a = $\dfrac{80}{16}$

⇒ a = 5.

G.P. is,

5, 5(2), 5(2)², 5(2)³…..

5, 10, 20, 40, …..

Hence, the G.P. is 5, 10, 20, 40, ……