    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

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

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B

$\frac{\sqrt{5}}{2}$

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C

$\sqrt{5}$

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D

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

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Solution

## The correct option is D $\frac{\left(\sqrt{5}-1\right)}{2}$The explanation for the correct optionLet 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, $a=ar+a{r}^{2}$.$⇒a=a\left(r+{r}^{2}\right)\phantom{\rule{0ex}{0ex}}⇒1=r+{r}^{2}\phantom{\rule{0ex}{0ex}}⇒{r}^{2}+r-1=0$The solution to the quadratic equation $a{x}^{2}+bx+c=0$ can be given by, $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$.Therefore, the solution for the equation ${r}^{2}+r-1=0$ is $r=\frac{-1±\sqrt{{\left(1\right)}^{2}-4\left(1\right)\left(-1\right)}}{2\left(1\right)}$.$⇒r=\frac{-1±\sqrt{1+4}}{2}\phantom{\rule{0ex}{0ex}}⇒r=\frac{-1±\sqrt{5}}{2}$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.  Suggest Corrections  7      Similar questions
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