In a G.P., the 4^th term is 48 and 7^th term is 384. Find its 6^th term.

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 G.P. = 48

⇒ a₄ = 48

⇒ ar^{4 - 1} = 48

⇒ ar³ = 48 ……….(1)

Given,

7^th term of G.P. = 384

⇒ a₇ = 48

⇒ ar^{7 - 1} = 384

⇒ ar⁶ = 384 ……….(2)

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

$\Rightarrow \dfrac{ar^6}{ar^3} = \dfrac{384}{48} \\[1em] \Rightarrow r^3 = 8 \\[1em] \Rightarrow r^3 = 2^3 \\[1em] \Rightarrow r = 2.$

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

⇒ a(2)³ = 48

⇒ a × 8 = 48

⇒ a = $\dfrac{48}{8}$ = 6.

⇒ a₆ = ar^{6 - 1}

⇒ a₆ = ar⁵

⇒ a₆ = 6 × 2⁵

⇒ a₆ = 6 × 32

⇒ a₆ = 192.

Hence, 6^th term of G.P. = 192.