Question

The first term of an infinite G.P. is x and its sum is 5. Then,

(a) – 10 < x < 0 

(b) 0 < x < 10 

(c) 0 ≤ x ≤ 10 

(d) x > 10

Answer 1

(b) 0 < \(x\) < 10 

Let the common ratio of the given G.P be r. Then,

\(S_\infty\) = \(\frac{x}{1-r}\) ⇒ 5 = \(\frac{x}{1-r}\) ⇒ 5 - 5r = \(x\)

∵ Sum to infinity of the given series is a finite quantity, | r | < 1.

∴ \(\bigg|1-\frac{x}{5}\bigg|\) < 1 ⇒ -1 \(\bigg(1-\frac{x}{5}\bigg)\)< 1

⇒ -1 < \(\bigg(\frac{x}{5}-1\bigg)\)< 1 (Multiplying the inequality by (– 1))

⇒ 0 < \(\frac{x}{5}\) < 2 ⇒ 0 < \(x\) < 10.