UPSC NDA Math Model Paper 6

Let's say that the GP is a, ar, ar‌2, ar3,....,arn−2,arn−1, with first term a and common ratio r.
Since the GP has only positive terms, it means that a > 0 and r > 0.
According to the question, arn=arn+1+arn+2, for any n. Dividing this equation by arn (which is > 0), we get:
⇒1=r+r2
⇒r2+r−1=0
⇒r=

−1±√12−4(1)(−1)

2

⇒r=

−1+√5

2

‌OR‌r=

−1−√5

2

Since, r is not negative (a > 0 and r > 0), the common ratio of the series is r=

−1+√5

2