Question

If every term of a $G.P$ with positive terms is the sum of its two previous terms, then common ratio of the series is

Answer 1

Finding the common ratio of the series:

Given that every term of a $G.P$ with positive terms is the sum of its two previous terms

Let $a,ar,a{r}^2$be the terms of the GP.

Given $a+ar=a{r}^2$

$r^2-r-1=0$

Solving using quadratic formula

We get $r=(1+\frac{\sqrt{5}}{2})or(1-\frac{\sqrt{5}}{2})$

Since GP consists of positive terms,

$r=(1+\frac{\sqrt{5}}{2})=(\frac{\sqrt{5}}{2}+1)$

Hence, the common root of the series is $\mathit{r}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{1}\mathbf{+}\frac{\sqrt{\mathbf{5}}}{\mathbf{2}}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\frac{\sqrt{\mathbf{5}}}{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}$