(A) The given $G.P.$ is $x^{3}, x^{5}, x^{7}, \ldots$ Here,the first term $a = x^{3}$ and the common ratio $r = \frac{x^{5}}{x^{3}} = x^{2}$. The sum

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$\frac{x^{3}(1-x^{2n})}{1-x^{2}}$

(A) The given $G.P.$ is $x^{3}, x^{5}, x^{7}, \ldots$ Here,the first term $a = x^{3}$ and the common ratio $r = \frac{x^{5}}{x^{3}} = x^{2}$. The sum of $n$ terms of a $G.P.$ is given by the formula $S_{n} = \frac{a(1-r^{n})}{1-r}$. Substituting the values of $a$ and $r$: $S_{n} = \frac{x^{3}(1-(x^{2})^{n})}{1-x^{2}}$ $S_{n} = \frac{x^{3}(1-x^{2n})}{1-x^{2}}$

Class 11 Mathematics Sequences and Series Geometric progression

Easy

Find the sum to the indicated number of terms in the geometric progression: $x^{3}, x^{5}, x^{7}, \ldots$ for $n$ terms (where $x \neq \pm 1$).

A
$\frac{x^{3}(1-x^{2n})}{1-x^{2}}$
B
$\frac{x^{3}(1-x^{n})}{1-x^{2}}$
C
$\frac{x^{2}(1-x^{2n})}{1-x^{2}}$
D
$\frac{x^{3}(1-x^{2n})}{1-x}$

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Sequences and Series

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Geometric progression

The $4^{\text{th}}$ term of a $GP$ is $500$ and its common ratio is $\frac{1}{m}$,where $m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this $GP$. If $S_6 > S_5+1$ and $S_7 < S_6+\frac{1}{2}$,then the number of possible values of $m$ is $..........$

DifficultJEE MAIN 2023

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Find four numbers forming a geometric progression in which the third term is greater than the first term by $9,$ and the second term is greater than the $4^{\text{th}}$ term by $18.$

Medium

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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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Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$,then $a_6$ is equal to

DifficultJEE MAIN 2025

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The terms of a $G.P.$ are positive. If each term is equal to the sum of the two terms that follow it,then the common ratio is:

Medium

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Find the sum to the indicated number of terms in the geometric progression: $x^{3}, x^{5}, x^{7}, \ldots$ for $n$ terms (where $x \neq \pm 1$).

Similar Questions

The $4^{\text{th}}$ term of a $GP$ is $500$ and its common ratio is $\frac{1}{m}$,where $m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this $GP$. If $S_6 > S_5+1$ and $S_7 < S_6+\frac{1}{2}$,then the number of possible values of $m$ is $..........$

Find four numbers forming a geometric progression in which the third term is greater than the first term by $9,$ and the second term is greater than the $4^{\text{th}}$ term by $18.$

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:

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$,then $a_6$ is equal to

The terms of a $G.P.$ are positive. If each term is equal to the sum of the two terms that follow it,then the common ratio is:

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