Question

The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is ___.

• A. $\frac{\sqrt{5}-1}{2}$
• B. $\frac{-1-\sqrt{5}}{2}$
• C. 1
• D. $2\sqrt{5}$

Answer 1

The correct option is A

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

The given condition can be expressed as
$T_n$ = $T_{n+1}+T_{n+2},$ where $n\ge 1.$
If $a$ is the first term and $r$ is the common ratio, then by the condition is
$a{r}^{n-1}=a{r}^n+a{r}^{n+1}.\Rightarrow r^{n-1}=r^n(1+r)$
$\Rightarrow r^2+r-1=0$
$\Rightarrow r=\frac{-1\pm \sqrt{1+4}}{2}=\frac{-1\pm \sqrt{5}}{2}$

Since each term is positive, the common ratio of the GP is $\frac{\sqrt{5}-1}{2}.$