Question

In a GP of positive terms, if any term is equal to the sum of next two terms, find the common ratio of the GP.

• A. $2\cos 9^o$
• B. $\sin {17}^o$
• C. $2\sin {18}^o$
• D. $\cos {18}^o$

Answer 1

Answer: (C)

$2\sin {18}^o$

Let a be the first term and r be the common ratio of the GP.

By hypothesis,

$a_n=a_{n+1}+a_{n+2}$

$a{r}^{n-1}=a{r}^n+a{r}^{n+1}$

$1=r+r^2$

$r^2+r-1=0$

$r=\frac{-1\pm \sqrt{1+4}}{2}=\frac{-1\pm \sqrt{5}}{2}$

But $r>0$. Therefore,

$r=\frac{-1+\sqrt{5}}{2}=2\left(\frac{\sqrt{5}-1}{4}\right)=2\sin {18}^o$