The 5^th and the 8^th terms of a G.P. are 32 and 256 respectively. Find its first term and the common ratio.

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

By formula,

⇒ a_n = ar^{n - 1}

Given,

5^th term of G.P. = 32

⇒ a₅ = 32

⇒ ar^{5 - 1} = 32

⇒ ar⁴ = 32 ……….(1)

Given,

8^th term of G.P. = 256

⇒ a₈ = 256

⇒ ar^{8 - 1} = 256

⇒ ar⁷ = 256 ……….(2)

Dividing equation (2) by (1), we get :

$\Rightarrow \dfrac{ar^7}{ar^4} = \dfrac{256}{32} \\[1em] \Rightarrow r^3 = 8 \\[1em] \Rightarrow r^3 = 2^3 \\[1em] \Rightarrow r = 2.$

Substituting value of r in equation (1), we get :

⇒ a × (2)⁴ = 32

⇒ 16a = 32

⇒ a = $\dfrac{32}{16}$ = 2.

Hence, first term = 2 and common ratio = 2.