The fourth term of a Geometric Progression (G.P.) is 16 and its seventh term is 128. Find its:

(a) We know that,

a_n = ar^{n - 1}

Given,

a₄ = ar³ = 16 ....(1)

a₇ = ar⁶ = 128 .....(2)

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

$$\begin{aligned}\Rightarrow \frac{ar^6}{ar^3} = \frac{128}{16} \\[1em] \Rightarrow r^3 = 8 \\[1em] \Rightarrow r = \sqrt[3]{8} = 2.\end{aligned}$$

Hence, common ratio = 2.

(b) The first term,

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

⇒ a(2)³ = 16

⇒ a(8) = 16

⇒ a = $\frac{16}{8}$

⇒ a = 2.

Hence, first term = 2.