In a GP series consisting of positive terms, | vedclass

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(B) Let $a_{n}$ be the general term of a $GP$ whose first term is $a$ and common ratio is $r$. According to the problem, each term is equal to the sum

Vedclass · Accepted Answer

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

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(B) Let $a_{n}$ be the general term of a $GP$ whose first term is $a$ and common ratio is $r$. According to the problem, each term is equal to the sum of the next two terms:$a_{n} = a_{n+1} + a_{n+2}$$a r^{n-1} = a r^{n} + a r^{n+1}$Since the terms are positive, $a 
eq 0$ and $r > 0$. Dividing by $a r^{n-1}$:$1 = r + r^{2}$$r^{2} + r - 1 = 0$Using the quadratic formula $r = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$:$r = \frac{-1 \pm \sqrt{1^{2} - 4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{5}}{2}$Since the $GP$ consists of positive terms, the common ratio $r$ must be positive.Therefore, $r = \frac{\sqrt{5}-1}{2}$.

In a $GP$ series consisting of positive terms, each term is equal to the sum of the next two terms. Then, the common ratio of this $GP$ series is

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