Question

Let $G_1$ and $G_2$ be the geometric means of two series $x_1,x_2,\dots \dots ,x_n$ and $y_1,y_2\dots .y_n$ respectively. If $G$ is the geometric mean of series $x_i/y_i,i=1,2,\dots ,n$, 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)

$\because G_1=(x_1{x}_2\dots x_n{)}^{1/n}$
and $G_2=(y_1{y}_2\dots y_n{)}^{1/n}$
$\therefore G={\left(\frac{x_1}{y_1}\times \frac{x_2}{y_2}\times \dots ..\times \frac{x_n}{y_n}\right)}^{1/n}$
$=\frac{{\left(x_1\times x_2\times \dots \dots \times x_n\right)}^{1/n}}{{\left(y_1\times y_2\times \dots \dots \times y_n\right)}^{1/n}}=\frac{G_1}{G_2}$