If the 5th term of a G.P. is 2, then the product of its first nine terms is :

1. 256

2. 1024

3. 512

4. 2048

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

Given,

5th term of a G.P. is 2.

⇒ T₅ = 2

⇒ ar^{5 - 1} = 2

⇒ ar⁴ = 2 …..(1)

Product of first 9 terms = a × ar¹ × ar² × …… × ar⁸

⇒ a⁹.r^{0 + 1 + 2 + …. + 8}

⇒ a⁹.r³⁶

⇒ (ar⁴)⁹ …..(2)

Substituting value of ar⁴ from equation (1) in (2), we get :

⇒ (2)⁹

⇒ 512.

Hence, option 3 is the correct option.