Question

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.

Options

• (–∞, –9] ≈ [3, ∞]

• [–3, ∞)

• (–∞, 9]

• (–∞, –3] ≈ [9, ∞]

Answer 1

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in (–∞, –3] ≈ [9, ∞].

Explanation:

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

Product = 27

⇒ `a/r.a.ar` = 27

⇒ a = 3

Sum = S  ...(Given)

⇒ `a/r + a + ar` = S

⇒ `3/r + 3 + 3r` = S

⇒ `3/r + 3r` = S – 3

⇒ 3r² – (S – 3)r + 3 = 0

For real r,

b² – 4ac ≥ 0

(S – 3)² – 4 × 3 × 3 ≥ 0

S² – 6r + 9 – 36 ≥ 0

S² – 6r – 27 ≥ 0

(S – 9)(S + 3) ≥ 0

⇒ –∞ < S ≥ –3

or 9 ≤ S < ∞