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 $G.P.$Given that every term is the sum of its two previous terms,we have the relatio

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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 $G.P.$Given that every term is the sum of its two previous terms,we have the relation: $T_n = T_{n-1} + T_{n-2}$.Substituting the general term formula $T_n = ar^{n-1}$,we get: $ar^{n-1} = ar^{n-2} + ar^{n-3}$.Dividing both sides by $ar^{n-3}$ (since $a > 0$ and $r > 0$),we obtain: $r^2 = r + 1$.Rearranging the terms gives the quadratic equation: $r^2 - r - 1 = 0$.Using the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$,we get $r = \frac{1 \pm \sqrt{1 - 4(1)(-1)}}{2} = \frac{1 \pm \sqrt{5}}{2}$.Since the terms of the $G.P.$ are positive,the common ratio $r$ must be positive. Therefore,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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