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

Under condition $T_n=T_{n+1}+T_{n+2}$
$\Rightarrow a{r}^{n-1}=a{r}^n+a{r}^{n+1}\Rightarrow r^{n-1}=r^n(1+r)$
$\Rightarrow r^2+r-1=0\Rightarrow r=\frac{-1\pm \sqrt{1+4}}{2}=\frac{-1\pm \sqrt{5}}{2}$
Since, each term is +ve.
Hence common ratio is $\frac{\sqrt{5}-1}{2}.$