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Q.
In a G.P. of alternating positive and negative terms, any term is the A.M. of the next two terms. Then the common ratio is
Sequences and Series
Solution:
Let the G.P. be $a-a r+a r^{2}-a r^{3}+\ldots$ with common ratio $=-r$
By the given condition
$a=\frac{a r^{2}-a r}{2} $
$\Rightarrow 2 a=a r^{2}-a r$
$\Rightarrow 2=r^{2}-r$
$\Rightarrow r^{2}-r-2=0$
$\Rightarrow (r-2)(r+1)=0$
$\Rightarrow r=2,-1$
$\therefore $ Common ratio $=-2$ or $1$
Hence, common ratio $=-2$
$(\because$ common ratio is $- ve )$