Question

Let $G_1$ and $G_2$ be the geometric means of two series $x _1, x _2, \ldots \ldots, x _{ n }$ and $y _1, y _2 \ldots . y _{ n }$ respectively. If $G$ is the geometric mean of series $x _{ i } / y _{ i }, i =1,2, \ldots, n$, 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

Answer: (D)

$\because G_1=(x_1 \,\,\, x_2 \,\,\, \ldots \,\,\, x_n)^{1 / n}$
and $G_2=(y_1 \,\,\, y_2 \,\,\, \ldots\,\,\,y_n)^{1 / n}$
$\therefore G=\left(\frac{x_1}{y_1} \times \frac{x_2}{y_2} \times \ldots . . \times \frac{x_n}{y_n}\right)^{1 / n}$
$=\frac{\left(x_1 \times x_2 \times \ldots \ldots \times x_n\right)^{1 / n}}{\left(y_1 \times y_2 \times \ldots \ldots \times y_n\right)^{1 / n}}=\frac{G_1}{G_2}$