Question

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of the progression is ____________.

Answer 1

Let a, ar, ar², ....... ar^n–1 defines a G.P
Since a = ar + ar² (given condition)
$$\begin{gathered} \mathrm{i}.\mathrm{e}1=r+r^2 \\ \mathrm{i}.\mathrm{e}{r}^2+r-1=0 \\ \mathrm{i}.\mathrm{e}r=\pm \frac{\sqrt{5}-1}{2}D=1-4\left(-1\right)=5 \end{gathered}$$
Since series is given to be positive
$\Rightarrow r=\frac{\sqrt{5}-1}{2}$
i.e the common ratio is $\frac{\sqrt{5}-1}{2}$