Question

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

Answer 1

Let a be the first term and r be the common ratio of the G.P. By hypothesis,

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

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

$1=r+r^2$

$r^2+r-1=0$

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

But, $r>0$. Therefore, $r=\frac{-1+\sqrt{5}}{2}=2\left(\frac{\sqrt{5}-1}{4}\right)=2\sin {18}^o$