Question

Every term of G.P. is positive and also every term is the sum of two preceding terms. Then, the common ratio of the G.P. is

• A. \(\frac{1 - \sqrt{5}}{2}\)
• B. \(\frac{\sqrt{5} + 1}{2}\)
• C. \(\frac{\sqrt{5} - 1}{2}\)
• D. 1

Answer 1

The Correct Option is B

Let \(a, ar, ar^2, \ldots\) be G.P. terms. Given: \[ ar^2 = ar + a \Rightarrow r^2 = r + 1 \Rightarrow r^2 - r - 1 = 0 \Rightarrow r = \frac{1 \pm \sqrt{5}}{2} \] Since all terms are positive, choose positive root: \[ r = \frac{1 + \sqrt{5}}{2} \]