Question

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

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

Answer 1

Answer: (B)

$\frac{1}{2}(\sqrt{5}-1)$

Let $a,ar,a{r}^2$ be the terms of G.P

$a=ar+a{r}^2$....[Given]

$\Rightarrow r^2+r-1=0$

We have a formula for solving quadratic equation $a{x}^2+bx+c=0$ is

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\Rightarrow r=\frac{-1\pm \sqrt{5}}{2}$

a negative value will be discarded because the terms in the question are positive.

$\therefore r=\frac{-1+\sqrt{5}}{2}$