Question

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 :-

• A. $\log G_1+\log G_2$
• B. $n_1\log G_1+n_2\log G_2$
• C. $\frac{\log G_1+\log G_2}{n_1+n_2}$
• D. $\frac{n_1\log G_1+n_2\log G_2}{n_1+n_2}$

Answer 1

Answer: (D)

Let $x_1,x_2,\dots x_{n_1}$ and $y_1,y_2,\dots .y_{n_2}$ are two series of size $n_1$ and $n_2$ resp.
$G_1={\left(x_1\times x_2\times \dots \times x_{n_1}\right)}^{1/n_1}.....$(1)
$G_2={\left(y_1\times y_2\times \dots \times y_{n_2}\right)}^{1/n_2}....$(2)
and $G={\left[\left(x_1\times x_2\times \dots \times x_{n_1}\right)\times \left(y_1\times y_2\times \dots y_{n_2}\right)\right]}^{\frac{1}{n_1+n_2}}$
$G={\left(G_1^{n_1}\times G_2^{n_2}\right)}^{1/n_1+n_2}$ [from (1) & (2)]
$\therefore \log G=\frac{n_1\log G_1+n_2\log G_2}{n_1+n_2}$