If $G_{1} a n d G_{2}$ are geometric mean of two series of sizes $n_{1} a n d n_{2}$ respectively and $G$ is geometric mean of their combined series, then log $G$ is equal to

log G_{1} + log G_{2}

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

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

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

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Solution

The correct option is D $\frac{n_{1} log G_{1} + n_{2} log G_{2}}{n_{1} + n_{2}}$
Let $x_{1}, x_{2}, . . . . x_{n_{1}}$ and $y_{1}, y_{2}, . . . . y_{n_{2}}$ are two series of size $n_{1}$ and $n_{2}$ respectively .
Geometric mean $G_{1} = {(x_{1} \times x_{2} \times . . . \times x_{n_{1}})}_{1}^{1 / n_{1}}$
$\Rightarrow G_{1}^{n_{1}} = x_{1} \times x_{2} \times . . . \times x_{n_{1}}$ ....(1)
$G_{2} = {(y_{1} \times y_{2} \times . . . . \times y_{n_{2}})}_{1}^{1 / n_{2}}$
$\Rightarrow G_{2}^{n_{2}} = y_{1} \times y_{2} \times . . . \times y_{n_{2}}$ .....(2)
Now, total number of terms in combined series will be $n_{1} + n_{2}$
$G = {[(x_{1} \times x_{2} \times . . . \times x_{n_{1}}) \times (y_{1} \times y_{2} \times . . . y_{n_{2}})]}^{\frac{1}{n_{1} + n_{2}}}$
$G = {(G_{1}^{n_{1}} \times G_{2}^{n_{2}})}^{1 / n_{1} + n_{2}}$ [by (1)&(2)]
$log G = \frac{1}{n_{1} + n_{2}} log (G_{1}^{n_{1}} \times G_{2}^{n_{2}})$
$∴ log G = \frac{n_{1} log G_{1} + n_{2} log G_{2}}{n_{1} + n_{2}}$

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Similar questions

Q. 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:

If $x_{1}, x_{2}, x_{3}, . . . .$ are n values of the variable x, then mathematical averages, Arithmetic Mean (A.M.), Geometric Mean (G.M) and Harmonic Mean (H.M) are count by the following formulas's.

$A . M . = \frac{x_{1} + x_{2} + x_{3} + . . . + x_{n}}{n} = \frac{1}{n} {n \sum i = 1 x_{1}} ≮ x_{1} > 0$

$G . M . = {(x_{1} \cdot x_{2} . . . x_{n})}_{1}^{1 / n}$ if each $x_{1} (i = 1, 2, . . . ., n)$ is positive and

$H . M = \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + . . . + \frac{1}{x_{n}}} = \frac{1}{\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{x_{1}})}$

* In case of frequency distribution $x / f_{1} (i = 1, 2, . . ., n)$ where $f_{1}$ is the frequency of the variable $x_{r}$ then the calculation of A.M.is counted as $A . M = n \sum i = 1 f_{1} x_{1} / N w h e r e N = f_{1} + f_{2} . . . . + f_{n}$

** If $w_{1}, w_{2}, . . . ., w_{_{n}}$ be the weight assigned to the n values

$x_{1}, x_{2}, . . . ., x_{_{n}}$ then weighted A.M is counted by $\frac{\sum_{i = 1}^{n} w_{1} x_{1}}{\sum_{i = 1}^{n} w_{1}}$

*** if $G_{1}, G_{2}$ are the G. M's of two series of sizes $n_{1}$ & $n_{2}$ respectively, then the geometric mean (G M.) of the combined series is counted by $log (G . M) = \frac{n_{1} log G_{1} + n_{2} log G_{2}}{n_{1} + n_{2}}$