Question

If in an infinite G.P, the first term equal to twice the sum of all the successive terms, then the common of this G.P. is

(a) \(\frac{1}{3}\) 

(b) \(\frac{1}{4}\)

(c) \(\frac{2}{5}\)

(d) \(\frac{2}{3}\)

Answer 1

(a) \(\frac{1}{3}\) 

Let the given G.P. be a, ar, ar², ar³, ..... , ∞. 

As the given infinite G.P. has a finite sum, | r | < 1 

Also, given a = 2 (ar + ar² + ar³ + .... + ∞)

⇒ a = 2\(\big(\frac{ar}{1-r}\big)\)  \(\big(\because\,S_\infty=\frac{a}{1-r}.\text{Here}\,a=ar, r=r\big)\)

⇒a – ar = 2ar 

⇒ 1 – r = 2r ⇒ 3r = 1 ⇒ r = \(\frac{1}{3}\).