Find the sum to indicated number of terms in each of the geometric progressions in $\left.x^{3}, x^{5}, x^{7}, \ldots n \text { terms (if } x \neq\pm 1\right)$
Find the sum to indicated number of terms in each of the geometric progressions in $\left.x^{3}, x^{5}, x^{7}, \ldots n \text { terms (if } x \neq\pm 1\right)$
The given $G.P.$ is $x^{3}, x^{5}, x^{7} \ldots .$
Here, $a=x^{3}$ and $r=x^{2}$
$S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}=\frac{x^{3}\left[1-\left(x^{2}\right)^{n}\right]}{1-x^{2}}=\frac{x^{3}\left(1-x^{2 n}\right)}{1-x^{2}}$
Similar Questions
Let $a_1, a_2, a_3, \ldots .$. be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, \ldots .$. be a sequence of positive integers in geometric progression with common ratio $2$ . If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality
$2\left(a_1+a_2+\ldots .+a_n\right)=b_1+b_2+\ldots . .+b_n$
holds for some positive integer $n$, is. . . . . . .
Let $a_1, a_2, a_3, \ldots .$. be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, \ldots .$. be a sequence of positive integers in geometric progression with common ratio $2$ . If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality
$2\left(a_1+a_2+\ldots .+a_n\right)=b_1+b_2+\ldots . .+b_n$
holds for some positive integer $n$, is. . . . . . .
- [IIT 2020]
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