If $G_{1}$ and $G_{2}$ are geometric means of two series of sizes $n_{1}$ and $n_{2}$ and if $G$ is geometric mean of combined series, then $log G$ is:

log G_{1} + log G_{2}

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\frac{n_{1} log G_{1} + n_{2} log G_{2}}{n_{1} + n_{2}}

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\frac{log G_{1} - log G_{2}}{n_{1} - n_{2}}

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none of these

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Solution

The correct option is B $\frac{n_{1} log G_{1} + n_{2} log G_{2}}{n_{1} + n_{2}}$
Geometric mean of $n_{1}$ items is $G_{1}$
Product of $n_{1}$ items is $G_{1}^{n_{1}}$
Geometric mean of $n_{2}$ items is $G_{2}$
Product of $n_{2}$ items is $G_{2}^{n_{2}}$
The product of the entire collection $= G_{1}^{n_{1}} . G_{2}^{n_{2}}$
$∴ G = (G_{1}^{n_{1}} . G_{2}^{n_{2}})^{\frac{1}{n_{1} + n_{2}}}$
(or) $G^{n_{1} + n_{2}} = G_{1}^{n_{1}} . G_{2}^{n_{2}}$
$(n_{1} + n_{2}) l o g G = l o g G_{1}^{n_{1}} + l o g G_{2}^{n_{2}}$
$(n_{1} + n_{2}) l o g G = n_{1} l o g G_{1} + n_{2} l o g G_{2}$
$l o g G = \frac{n_{1} l o g G_{1} + n_{2} l o g G_{2}}{n_{1} + n_{2}}$

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