Question

The 5^th and the 8^th terms of a G.P. are 32 and 256 respectively. Find its first term and the common ratio.

Answer 1

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

By formula,

⇒ a_n = ar^{n − 1}

Given,

5^th term of G.P. = 32

⇒ a₅ = 32

⇒ ar^{5 − 1} = 32

⇒ ar⁴ = 32 ..........(1)

Given,

8^th term of G.P. = 256

⇒ a₈ = 256

⇒ ar^{8 − 1} = 256

⇒ ar⁷ = 256 ..........(2)

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

⇒ `(ar^7)/(ar^4) = 256/32` 

⇒ r³ = 8

⇒ r³ = 23

⇒ r = 2

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

⇒ a × (2)⁴ = 32

⇒ 16a = 32

⇒ a = `32/16`​

a = 2

Hence, first term = 2 and common ratio = 2.