Question

Let $G_{1}and{G}_2$ be the geometric means of two series $x_1,x_2,.....x_{n}and{y}_1,y_2.....,y_n$ respectively. If G is the geometric mean of series $x_i/y_i$, $i=1,2,....,n$, then G is equal to

• A. $G_1-G_2$
• B. $\log G_1/\log G_2$
• C. $\log \left(G_1/G_2\right)$
• D. $G_1/G_2$

Answer 1

The correct option is D $G_1/G_2$

Given series $x_1,x_2,.....x_n$.
Geometric mean $G_1={(x_1\times x_2\times ....x_n)}^{1/n}$ ....(1)
Another series $y_1,y_2.....,y_n$.
Geometric mean $G_2={(y_1\times y_2\times ....y_n)}^{1/n}$ ....(2)
Also given series= $\frac{x_1}{y_1}\frac{x_2}{y_2}....\frac{x_n}{y_n}$
Geometric mean $G={(\frac{x_1}{y_1}\times \frac{x_2}{y_2}\times \cdot \cdot \cdot \cdot \cdot \times \frac{x_n}{y_n})}^{1/n}$
$\Rightarrow G=\frac{{(x_1\times x_2\times x_3\cdot \cdot \cdot \times x_n)}^{1/n}}{{(y_1\times y_2\times y_3\cdot \cdot \cdot \times y_n)}^{1/n}}$
$\Rightarrow G=\frac{G_1}{G_2}$ (by (1)and (2))