A)

Given: 2, 2√2, 4….. and a_n = 128

Here, in the above G.P.

a = 2

Common ratio(r) =

We know that in G.P a_n = ar^n-1

∴ a_n = 2 × (√2)^n-1

⇒ 128 = 2 × (√2)^n-1

⇒ (√2)^n-1 = 128/2 = 64

⇒ = 64 (∵ √x = x^1/2)

Apply ln on both sides

We get

⇒

⇒

⇒

⇒ n – 1 = 12

⇒ n = 13

∴ a₁₃ = 128

B)

Given: √3, 3, 3√3 ….. and a_n = 729

Here, in the above G.P.

a = √3

Common ratio(r)

We know that in G.P a_n = ar^n-1

∴ a_n = √3 × (√3)^n-1

⇒ 729 = (√2)ⁿ

⇒ (√2)ⁿ = 729

⇒ = 729 (∵ √x = )

Apply ln on both sides

We get

⇒ n = 12

∴ a₁₂ = 729

C)

Given: and a_n =

Here, in the above G.P.

a =

Common ratio(r) = =

We know that in G.P a_n = ar^n-1

∴ a_n =

⇒

Apply ln on both sides

We get

⇒

⇒

⇒ n = 9

∴ a₉ =