Question

In a G.P. of positive terms, for a fixed ${}^{\prime }{n}^{\prime }$, the nth term is equal to 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. $\cos$${18}^{\circ }$
• D. $2\sin$${18}^{\circ }$

Answer 1

Answer: (D)

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

The nth term of a G.P. is $a{r}^{n-1}$
By the given condition we can write as $a{r}^{n-1}=a{r}^n+a{r}^{n+1}$
After simplifying we get a quadratic equation in r,
$r^2+r-1=0$
Solving for r we get 2 values of r: one positive and other negative.
Only positive one is considered as the terms are all positive (given).
So r $=\frac{\sqrt{5}-1}{2}$, which is the value of 2 $\sin {18}^{\circ }$