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

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