Question

In a geometric progression of all positive terms, any term is equal to the sum of its next two terms. Then the common ratio of this progression is

• A. $2\cos {18}^{\circ }$
• B. $\sin {18}^{\circ }$
• C. $\cos {18}^{\circ }$
• D. $2\sin {18}^{\circ }$

Answer 1

The correct option is D

$2\sin {18}^{\circ }$

Let the sequence be $a,ar,a{r}^2,...,a{r}^k,a{r}^{k+1},a{r}^{k+2}$, …

Given condition is $a{r}^k=a{r}^{k+1}+a{r}^{k+2}$

As $a\ne 0$ and $r\ne 0$, hence

$1=r+r^2$

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

As $r$ must be positive,

$\Rightarrow r=\frac{-1+\sqrt{5}}{2}=2\sin {18}^{\circ }$