If the 4th and 7th terms of a G.P. are 54 and 1458 respectively, find the G.P.

a₄ = 54 and a₇ = 1458

∵ a_n = a₁r^n-1 (n = no. of term, a₁ = first term of G.P, r = common ratio)

⇒ a₄ = a₁r³

⇒ 54 = a₁r³ ………(1)

Also, ⇒ a₇ = a₁r⁶

⇒ 1458 = a₁r⁶ ………(2)

Dividing equation (1) & (2), we get-

⇒ r³ = 27

⇒ r³ = 3³

⇒ r = 3

Now, putting value of r in equation (1), we get-

⇒54 = a₁3³

⇒ a₁ = 2

Now, G.P will be-

⇒ a₁ , a₁r ,a₁r², a₁r³,………

⇒ 2, 2 × 3, 2 × 3², 2 × 3…

⇒ 2, 6, 18, 54,… is the G.P.