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# If G be the G.M. of the product of K, Set of observations with G.M.’S ${{G}_{1}},{{G}_{2}},{{G}_{3}},....,{{G}_{K}}$ Respectively, then G equals.A) $\log {{G}_{1}}+\log {{G}_{2}}+\log {{G}_{3}}+...+\log {{G}_{k}}$ B) ${{G}_{1}}{{G}_{2}}{{G}_{3}}...{{G}_{k}}$ C) $\log {{G}_{1}}.\log {{G}_{2}}.\log {{G}_{3}}...\log {{G}_{k}}$ D) $k\log \left( {{G}_{1}}.{{G}_{2}}.{{G}_{3}}...{{G}_{k}} \right)$

Last updated date: 17th Jun 2024
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Hint: We will assume $k$ variables and then find the geometric mean of the variables. For that equation apply logarithmic both sides and use the formula $\log {{a}^{b}}=b\log a$ and then substitute the product of the $k$ variables. Now use the formula $\log \left( ab \right)=\log a+\log b$ to get the result.

If ${{x}_{1}},{{x}_{2}},{{x}_{3}},...{{x}_{k}}$ be the $k$ variables and their product is denoted by $x={{x}_{1}}.{{x}_{2}}.{{x}_{3}}...{{x}_{k}}$
The geometric mean of the $k$ variables is $G={{\left( x \right)}^{\dfrac{1}{k}}}$
Take $\log$ equation on both sides we have
$\Rightarrow$ $\log G=\log {{\left( x \right)}^{\dfrac{1}{k}}}$
Using the formula $\log {{a}^{b}}=b\log a$ in the above equation, then
$\Rightarrow$ $\log G=\dfrac{1}{k}\log \left( x \right)$
Substitute $x={{x}_{1}}.{{x}_{2}}.{{x}_{3}}...{{x}_{k}}$ in the above equation we get
$\Rightarrow$ $\log G=\dfrac{1}{k}\log \left( {{x}_{1}}.{{x}_{2}}.{{x}_{3}}...{{x}_{k}} \right)$
Using the formula $\log \left( ab \right)=\log a+\log b$ in the above equation, then
\begin{align} & \log G=\dfrac{1}{k}\log {{x}_{1}}+\dfrac{1}{k}\log {{x}_{2}}+\dfrac{1}{k}\log {{x}_{3}}+...+\dfrac{1}{k}\log {{x}_{k}} \\ & \log G=\log {{G}_{1}}+\log {{G}_{2}}+\log {{G}_{3}}+...+\log {{G}_{k}} \\ & \log G=\log \left( {{G}_{1}}{{G}_{2}}{{G}_{3}}...{{G}_{k}} \right) \\ & G={{G}_{1}}{{G}_{2}}{{G}_{3}}...{{G}_{k}} \end{align}

Note:
Please note that we are using the proper logarithmic function at the right place in order to get the result. Some of other logarithmic functions are
\begin{align} & \log a-\log b=\log \left( \frac{a}{b} \right) \\ & \log \left( \frac{1}{y} \right)=\log \left( {{y}^{-1}} \right)=-\log y \\ & {{\log }_{a}}a=1 \\ & {{\log }_{a}}\left( {{a}^{b}} \right)=b \\ & {{a}^{{{\log }_{a}}\left( b \right)}}=b \end{align}
Geometric Sequence: In a sequence if the numbers are obtained by multiplying a constant with the previous number (except first number) then that sequence is called a Geometric sequence.
We can write the general form of Geometric Sequence as $a,ar,a{{r}^{2}},a{{r}^{3}},...$
Where $a$ is the first term and
$r$ is the constant value that is multiplied to the previous term.
Ex: $1,2,4,8,16,...$ . Here you can find that each term (except the first term) is obtained by multiplying a constant value $\left( 2 \right)$ to the previous term. Here we can write $a=1$ and $r=2$
Geometric Mean: The geometric mean is the special type of average and calculated as ${{n}^{th}}$ root of the product of $n$ values. Mathematically geometric mean of the series $a,{{a}_{1}},{{a}_{2}}$ is
$G.M=\sqrt[3]{a\left( {{a}_{1}} \right)\left( {{a}_{2}} \right)}$