If each term of a geometric progression (GP) is positive and is the sum of two preceding terms, then the common ratio of the GP is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\frac{\sqrt{5}+1}{2}\)
In a GP each term is the one before it multiplied by the common ratio \(r\), so three terms in a row can be written \(a,\ ar,\ ar^2\). The condition that each term is the sum of the two terms before it gives an equation in \(r\).
Third term \(=\) sum of the two before it: \(ar^2=a+ar\)
As \(a\neq0\) (the terms are positive): \(r^2=1+r\), that is, \(r^2-r-1=0\)
By the quadratic formula: \(\displaystyle r=\frac{1\pm\sqrt{1+4}}{2}=\frac{1\pm\sqrt{5}}{2}\)
All the terms are positive, so \(r\) must be positive; \(\frac{1-\sqrt{5}}{2}\) is negative, so \(\displaystyle r=\frac{\sqrt{5}+1}{2}\)
Tip: Option (1), \(\frac{\sqrt{5}-1}{2}\), is the answer when each term is the sum of the two terms after it (\(a=ar+ar^2\)). Read whether the question says preceding or following.