If $g_{1} & g_{2}$ be the geometric means of two series of $n_{1} & n_{2}$ items.Then the G.M.of the series obtained on combining them is

{[{(g_{1})}_{1}^{n_{1}} . {(g_{2})}_{2}^{n_{2}}]}^{\frac{1}{n_{1} + n_{2}}}

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{(g_{1} g_{2})}_{1}^{\frac{n_{1}}{n_{1} + n_{2}}}

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{(g_{1} g_{2})}_{1}^{\frac{n_{2}}{n_{1} + n_{2}}}

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{(g_{1} g_{2})}_{1}^{\frac{n_{1} n_{2}}{n_{1} + n_{2}}}

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Solution

The correct option is A ${[{(g_{1})}_{1}^{n_{1}} . {(g_{2})}_{2}^{n_{2}}]}^{\frac{1}{n_{1} + n_{2}}}$
Let $x_{1}, x_{2}, . . . . x_{n_{1}}$ and $y_{1}, y_{2}, . . . . y_{n_{2}}$ be two series of sizes $n_{1}$ and $n_{2}$ respectively.
Now, $g_{1} = {(x_{1} x_{2} \dots x_{n_{1}})}_{1}^{\frac{1}{n_{1}}}$
$\Rightarrow g_{1}^{n} = x_{1} x_{2} . . . . x_{n}$ ....(1)
$g_{2} = {(y_{2} y_{2} \dots y_{n_{2}})}_{2}^{\frac{1}{n_{2}}}$
$\Rightarrow g_{2}^{n} = y_{1} y_{2} . . . . y_{n}$ .....(2)
Total number of terms in combined series $= n_{1} + n_{2}$
If $g$ is the G.M.of the combined series $g = {(x_{1} x_{2} \dots x_{n_{1}}) \times (y_{1} y_{2} \dots y_{n_{2}})}^{\frac{1}{n_{1} + n_{2}}}$
$g = {g_{1}^{n_{1}} . g_{2}^{n_{2}}}^{\frac{1}{n_{1} + n_{2}}}$

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