(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 = \(\frac{1}{3}\).