(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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Class 11 Questions

Class 11

Mathematics Questions

Chapter

Sequences and Series

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Geometric progression

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

DifficultJEE MAIN 2021

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For a sequence $(t_n)$,if $S_n = 5(2^n - 1)$,then $t_n = \ldots$

EasyMHT CET 2019

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The sum of the first two terms of a geometric progression is $12$. The sum of the third and fourth terms is $48$. The terms of the geometric progression alternate between positive and negative. What is the first term?

Difficult

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In a geometric progression with $n$ terms and a common ratio of $3$,the sum of the $n$ terms is $364$ and the last term is $243$. Find the value of $n$.

Medium

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The sum of some terms of a $G.P.$ is $315$,whose first term and the common ratio are $5$ and $2$ respectively. Find the last term and the number of terms.

Medium

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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

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

For a sequence $(t_n)$,if $S_n = 5(2^n - 1)$,then $t_n = \ldots$

The sum of the first two terms of a geometric progression is $12$. The sum of the third and fourth terms is $48$. The terms of the geometric progression alternate between positive and negative. What is the first term?

In a geometric progression with $n$ terms and a common ratio of $3$,the sum of the $n$ terms is $364$ and the last term is $243$. Find the value of $n$.

The sum of some terms of a $G.P.$ is $315$,whose first term and the common ratio are $5$ and $2$ respectively. Find the last term and the number of terms.

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