The 4^th term of a G.P. is 54 and its 7^th term is 1458, the common ratio of this G.P. is :

1. $\dfrac{1}{3}$

2. 3

3. -3

4. $-\dfrac{1}{3}$

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

By formula,

⇒ a_n = ar^{n - 1}

Given,

4^th term of a G.P. is 54.

⇒ a₄ = 54

⇒ ar^{4 - 1} = 54

⇒ ar³ = 54 ………(1)

7^th term of a G.P. is 1458.

⇒ a₇ = 1458

⇒ ar^{7 - 1} = 1458

⇒ ar⁶ = 1458 ………(2)

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

$\Rightarrow \dfrac{ar^6}{ar^3} = \dfrac{1458}{54} \\[1em] \Rightarrow r^3 = 27 \\[1em] \Rightarrow r^3 = 3^3 \\[1em] \Rightarrow r = 3.$

Hence, Option 2 is the correct option.