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 equals

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

Answer 1

The correct option is D

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

The explanation for the correct option

Let us assume that the first term of the geometric progression is $a$ and the common ratio is $r$.

Thus, the first three terms of the progression can be given by $a$, $ar$ and $a{r}^2$.

It is given that each term equals the sum of the next two terms.

Thus, $a=ar+a{r}^2$.

$$\begin{gathered} \Rightarrow a=a\left(r+r^2\right) \\ \Rightarrow 1=r+r^2 \\ \Rightarrow r^2+r-1=0 \end{gathered}$$

The solution to the quadratic equation $a{x}^2+bx+c=0$ can be given by, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

Therefore, the solution for the equation $r^2+r-1=0$ is $r=\frac{-1\pm \sqrt{{\left(1\right)}^2-4\left(1\right)\left(-1\right)}}{2\left(1\right)}$.

$$\begin{gathered} \Rightarrow r=\frac{-1\pm \sqrt{1+4}}{2} \\ \Rightarrow r=\frac{-1\pm \sqrt{5}}{2} \end{gathered}$$

As the progression consists only of positive terms, therefore, the common ratio of the progression, $r=\frac{\sqrt{5}-1}{2}$.

Hence, (D) is the correct option.