# In a geometric progression consisting of positive terms, each term equals the sum of the next two terms.

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 ____________.

Solution:

Let a, ar, ar2, ....... arn–1 defines a G.P

Since a = ar + ar2         (given condition)

i. e $1=r+r^{2} i. e$r^{2}+r-1=0$i. e$r=\pm \frac{\sqrt{5}-1}{2} \quad D=1-4(-1)=5$Since series is given to be positive$\Rightarrow r=\frac{\sqrt{5}-1}{2}$i.e the common ratio is$\frac{\sqrt{5}-1}{2}\$