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  4. In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is

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Q. In a $G.P$. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the $G.P$. is

Sequences and Series

Solution:

$t_{n} = t_{n+1} + t_{n+2} $
$\Rightarrow ar^{n-1}= ar^{n} +ar^{n+1} $
$\Rightarrow 1 = r+ r^{2} $
$ \Rightarrow r = \frac{-1\pm\sqrt{5}}{2}$, since $r > 0$
Therefore, $r= 2\left(\frac{\sqrt{5}- 1}{4}\right)$
$ = 2sin\,18^{\circ} $