In a geometric progression consisting of positive terms, each term equ

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

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»

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