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. $\begin{array}{l}\frac{1-\sqrt{5}}{2}\end{array}$
• B. $\begin{array}{l}\frac{\sqrt{5}}{2}\end{array}$
• C. $\sqrt{5}$
• D. $\begin{array}{l}\frac{\sqrt{5}-1}{2}\end{array}$

Answer 1

The correct option is D

$\begin{array}{l}\frac{\sqrt{5}-1}{2}\end{array}$

Explanation for the correct answer:

Find the common ratio of this progression.

Let, $a,ar,a{r}^2$ be three terms of GP.

Given,

$$\begin{gathered} a=ar+a{r}^2 \\ \Rightarrow 1=r+r^2 \\ \Rightarrow r+r^2-1=0 \\ \Rightarrow r=\frac{-1\pm \sqrt{1^2-4\left(1\right)\left(-1\right)}}{2\left(1\right)}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{quadratic}\mathbf{}\mathbf{formula}\mathbf{}\mathbf{=}\frac{\mathbf{}\mathbf{-}\mathbf{b}\mathbf{\pm }\sqrt{{\mathbf{b}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{a}\mathbf{c}}}{\mathbf{2}\mathbf{a}}\mathbf{}\mathbf{where}\mathbf{}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{,}\mathbf{c}\mathbf{}\mathbf{are}\mathbf{}\mathbf{coefficients}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{x}}^{\mathbf{2}}\mathbf{,}\mathbf{x}\mathbf{}\mathbf{and}\mathbf{}\mathbf{quadratic}\mathbf{}\mathbf{constant}\mathbf{}\mathbf{respectively}\mathbf{.}\mathbf{]} \\ \Rightarrow r=\frac{-1\pm \sqrt{5}}{2} \\ \Rightarrow r=\frac{-1+\sqrt{5}}{2}or\frac{-1-\sqrt{5}}{2} \end{gathered}$$

We only consider,

$$\begin{gathered} r=\frac{-1+\sqrt{5}}{2}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{given}\mathbf{,}\mathbf{}\mathbf{}\mathbf{geometric}\mathbf{}\mathbf{progression}\mathbf{}\mathbf{consisting}\mathbf{}\mathbf{of}\mathbf{}\mathbf{positive}\mathbf{}\mathbf{terms}\mathbf{]} \\ \therefore r=\frac{\sqrt{5}-1}{2} \end{gathered}$$

Hence, the correct option is (D).