Question

In a geometric progression consisting of positive terms, each term equals the sum of next two terms. Then, the common ratio of the progression equals

(a) \(\frac{\sqrt5}{2}\)

(b) \(\sqrt5\)

(c) \(\frac{\sqrt5-1}{2}\)

(d) \(\frac{\sqrt5+1}{2}\)

Answer 1

(c) \(\frac{\sqrt5-1}{2}\)

Let the G.P. be a, ar, ar², ...... 

As all the terms of the given G.P. are positive, a > 0, r > 0. 

Given, a = ar + ar² 

⇒ ar² + ar – a = 0 

⇒ r² + r – 1 = 0.

∴ r = \(\frac{-1\pm\sqrt{1-4}}{2}\) = \(\frac{-1\pm\sqrt{5}}{2}\)

⇒ r = \(\frac{\sqrt{5-1}}{2}\).            \(\big(\because\frac{-1-\sqrt5}{2}\,\text{is a negative quantity and }\,r>0\big)\)