Question

The sum of three terms (numbers) of a GP. is `3 1/2` and their product is 1; the numbers are ______.

Options

• `1/2`, 1 and 2

• `1/3`, 3 and 9

• 1, `1/2`, and 2

• 2, `1/2`, and 1

Answer 1

The sum of three terms (numbers) of a GP. is `3 1/2` and their product is 1; the numbers are `bbunderline(1/2, 1 and 2)`.

Explanation:

Let three terms of G.P. be `a/r`, a, ar

Given,

Product of three terms of G.P. = 1

∴ ar × a × ar = 1

⇒ a³ = 1

⇒ a³ = 13

⇒ a = 1

Given,

Sum of three terms of G.P. = 312

⇒ `a/r + a + ar = 3 1/2`

⇒ `1/r + 1 + 1(r) = 7/2   [∵ a = 1]`

⇒ `1/r ​+ 1 + r = 7/2​`

⇒ `(1+r+r^2​)/r = 7/2​`

⇒ 2(r² + r + 1) = 7r

⇒ 2r² + 2r + 2 = 7r

⇒ 2r² + 2r − 7r + 2 = 0

⇒ 2r² − 5r + 2 = 0

⇒ 2r² − 4r − r + 2 = 0

⇒ 2r(r − 2) − 1(r − 2) = 0

⇒ (2r − 1) (r − 2) = 0

⇒ 2r − 1 = 0 or r − 2 = 0

⇒ 2r = 1 or r = 2

⇒ r = `1/2`​ or r = 2

Let r = `1/2

Terms:

⇒ `a/r`, a, ar

⇒ `1/(1/2), 1, 1 × 1/2`

⇒ `2, 1, 1/2`

Let r = 2

Terms:

⇒ `a/r`, a, ar

⇒ `1/2, 1, 1 × 2`

⇒ `1/2, 1, 2`