If every term of a G.P. with positive terms i | vedclass

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(D) Let the first term be $a$ and the common ratio be $r$ for the $G.P.$ with positive terms.According to the given condition,every term is the sum of

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

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(D) Let the first term be $a$ and the common ratio be $r$ for the $G.P.$ with positive terms.According to the given condition,every term is the sum of its two previous terms:$T_n = T_{n-1} + T_{n-2}$Substituting the general term formula $T_n = ar^{n-1}$:$ar^{n-1} = ar^{n-2} + ar^{n-3}$Dividing both sides by $ar^{n-3}$ (since $a > 0$ and $r > 0$):$r^2 = r + 1$$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 terms are positive,the common ratio $r$ must be positive. Thus,we take the positive root:$r = \frac{1 + \sqrt{5}}{2}$

If every term of a $G.P.$ with positive terms is the sum of its two previous terms,then the common ratio of the series is

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