Question

In a G.P. of positive terms, if any term is equal to the sum of the next two terms, then the common ratio of the G.P. is

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

Answer 1

The correct option is C $2\sin {18}^{\circ }$

It is given that any term is the sum of next $2$ terms
$\Rightarrow t_n=t_{n+1}+t_{n+2}$
$\Rightarrow a{r}^{n-1}=a{r}^n+a{r}^{n+1}$
$\Rightarrow 1=r+r^2$
$\Rightarrow r^2+r-1=0$
$\Rightarrow (r-\frac{-1+\sqrt{5}}{2})(r-\frac{-1-\sqrt{5}}{2})=0$
$\Rightarrow r=\frac{-1\pm \sqrt{5}}{2}$
$\Rightarrow r=\frac{\sqrt{5}-1}{2}$
$\Rightarrow r=2\left(\frac{\sqrt{5}-1}{4}\right)$
$\Rightarrow r=2\sin {18}^{\circ }$
Hence the option (D) is correct.