The terms of a G.P. are positive. If each ter | vedclass

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(A) Let the terms of the $G.P.$ be $a, ar, ar^2, ar^3, \dots$ where $a > 0$ and $r > 0$.Given that each term is equal to the sum of the two terms that

Vedclass · Accepted Answer

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

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(A) Let the terms of the $G.P.$ be $a, ar, ar^2, ar^3, \dots$ where $a > 0$ and $r > 0$.Given that each term is equal to the sum of the two terms that follow it,we have:$T_n = T_{n+1} + T_{n+2}$$ar^{n-1} = ar^n + ar^{n+1}$Dividing both sides by $ar^{n-1}$ (since $a 
eq 0$ and $r 
eq 0$):$1 = r + r^2$$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 terms are positive,the common ratio $r$ must be positive.Therefore,$r = \frac{\sqrt{5} - 1}{2}$.

The terms of a $G.P.$ are positive. If each term is equal to the sum of the two terms that follow it,then the common ratio is:

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