(D) Given that $G_1 = (x_1 \times x_2 \times \dots \times x_n)^{1/n}$ and $G_2 = (y_1 \times y_2 \times \dots \times y_n)^{1/n}$.The geometric mean $G

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(D) Given that $G_1 = (x_1 \times x_2 \times \dots \times x_n)^{1/n}$ and $G_2 = (y_1 \times y_2 \times \dots \times y_n)^{1/n}$.The geometric mean $G$ of the series $\frac{x_i}{y_i}$ is defined as:$G = \left( \frac{x_1}{y_1} \times \frac{x_2}{y_2} \times \dots \times \frac{x_n}{y_n} \right)^{1/n}$$G = \frac{(x_1 \times x_2 \times \dots \times x_n)^{1/n}}{(y_1 \times y_2 \times \dots \times y_n)^{1/n}}$Substituting the values of $G_1$ and $G_2$:$G = \frac{G_1}{G_2}$

Class 11 Mathematics Sequences and Series Geometric progression

Easy

Let $G_1$ and $G_2$ be the geometric means of two series $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ respectively. If $G$ is the geometric mean of the series $\frac{x_i}{y_i}$ where $i = 1, 2, \dots, n$,then what is $G$ equal to?

A
$G_1 - G_2$
B
$\frac{\log G_1}{\log G_2}$
C
$\log (G_1/G_2)$
D
$G_1/G_2$

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Geometric progression

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

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Find the $20^{\text{th}}$ and $n^{\text{th}}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \dots$

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If $|\alpha| < 1$ and $|\beta| < 1$,and $1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty = s_1$ and $1 - \beta + \beta^2 - \beta^3 + \dots \infty = s_2$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals:

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If three successive terms of a $G.P.$ with common ratio $r$ $(r > 1)$ are the lengths of the sides of a triangle,and $[r]$ denotes the greatest integer less than or equal to $r$,then $3[r] + [-r]$ is equal to:

DifficultJEE MAIN 2024

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Let $a_{1}, a_{2}, a_{3}, \dots$ be a $G$.$P$. of increasing positive terms such that $a_{2} \cdot a_{3} \cdot a_{4} = 64$ and $a_{1} + a_{3} + a_{5} = \frac{813}{7}$. Then $a_{3} + a_{5} + a_{7}$ is equal to:

DifficultJEE MAIN 2026

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Let $G_1$ and $G_2$ be the geometric means of two series $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ respectively. If $G$ is the geometric mean of the series $\frac{x_i}{y_i}$ where $i = 1, 2, \dots, n$,then what is $G$ equal to?

Similar Questions

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

Find the $20^{\text{th}}$ and $n^{\text{th}}$ terms of the $G.P.$ $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \dots$

If $|\alpha| < 1$ and $|\beta| < 1$,and $1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty = s_1$ and $1 - \beta + \beta^2 - \beta^3 + \dots \infty = s_2$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals:

If three successive terms of a $G.P.$ with common ratio $r$ $(r > 1)$ are the lengths of the sides of a triangle,and $[r]$ denotes the greatest integer less than or equal to $r$,then $3[r] + [-r]$ is equal to:

Let $a_{1}, a_{2}, a_{3}, \dots$ be a $G$.$P$. of increasing positive terms such that $a_{2} \cdot a_{3} \cdot a_{4} = 64$ and $a_{1} + a_{3} + a_{5} = \frac{813}{7}$. Then $a_{3} + a_{5} + a_{7}$ is equal to:

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