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. $sin18^{\circ }$
• B. $2cos18^{\circ }$
• C. $cos18^{\circ }$
• D. $2sin18^{\circ }$

Answer 1

Answer: (D)

$t_n=t_{n+1}+t_{n+2}$
$\Rightarrow a{r}^{n-1}=a{r}^n+a{r}^{n+1}$
$\Rightarrow 1=r+r^2$
$\Rightarrow r=\frac{-1\pm \sqrt{5}}{2}$, since $r>0$
Therefore, $r=2\left(\frac{\sqrt{5}-1}{4}\right)$
$=2sin18^{\circ }$