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Q.
The product of 6 geometric means between 8 and $\frac{1}{16}$ will be:
Sequences and Series
Solution:
There are 6 G.M.s between 8 and $\frac{1}{16}$
$\therefore 8, G_1, G_2, \ldots \ldots, G_6, \frac{1}{16}$ are in G.P.
we know that product of $n$ G.M. between two numbers is $n^{\text {th }}$ power of single G.M. between two numbers.
G.M. of 8 and $\frac{1}{16}$ is $\sqrt{8 \cdot \frac{1}{16}}=\frac{1}{\sqrt{2}}$
$\therefore $ product of 6 G.Ms $=\left(\frac{1}{\sqrt{2}}\right)^6=\frac{1}{8}$