Thank you for reporting, we will resolve it shortly
Q.
If $G_1$ and $G_2$ are geometric mean of two series of sizes $n_1$ and $n_2$ resp. and $G$ is geometric mean of their combined series, then $\log G$ is equal to :-
Statistics
Solution:
Let $x _1, x _2, \ldots x _{ n _1}$ and $y _1, y _2, \ldots . y _{ n _2}$ are two series of size $n_1$ and $n_2$ resp.
$G_1=\left(x_1 \times x_2 \times \ldots \times x_{n_1}\right)^{1 / n_1}.....$(1)
$G_2=\left(y_1 \times y_2 \times \ldots \times y_{n_2}\right)^{1 / n_2} ....$(2)
and $ G=\left[\left(x_1 \times x_2 \times \ldots \times x_{n_1}\right) \times\left(y_1 \times y_2 \times \ldots y_{n_2}\right)\right]^{\frac{1}{n_1+n_2}}$
$G=\left(G_1^{ n _1} \times G_2^{ n _2}\right)^{1 / n _1+ n _2} $ [from (1) & (2)]
$\therefore \log G=\frac{ n _1 \log G_1+ n _2 \log G_2}{n_1+n_2}$