Question

In a GP of positive terms any term is equal to the sum of the next two terms The common ratio of the GP is

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

Answer 1

Answer: (D)

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

Let the $n$ terms of GP be $a,ar,a{r}^2.....a{r}^{n-1}$

now given $a=ar+a{r}^2$

so $1=r+r^2$

roots of this quadratic eqn are

$=\frac{-1\pm \sqrt{5}}{2}$

−1±52

we'll take only the positive value as all terms are positive

$r=\frac{\sqrt{5}-1}{2}$

we know that $sin{18}^{\circ }=\frac{\sqrt{5}-1}{4}$

so, $r=2sin{18}^{\circ }$