Question:
Find the sum of the GP :
$\mathbf{x}^{3}+\mathbf{x}^{5}+\mathbf{x}^{7}+\ldots$ To $\mathbf{n}$ terms
Solution:
Sum of a G.P. series is represented by the formula, $S_{n}=a \frac{r^{n}-1}{r-1}$ when r≠1. ‘Sn’ represents the sum of the G.P. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.
Here
$a=x^{3}$
$r=($ ratio between the $n$ term and $n-1$ term $) x^{5} \div x^{3}=x^{2}$
n terms
$\therefore S_{n}=x^{3} \times \frac{x^{2 n}-1}{x^{2}-1}$
$\Rightarrow \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{x}^{3}\left(\mathrm{x}^{\mathrm{n}}-1\right)\left(\mathrm{x}^{\mathrm{n}}+1\right)}{(\mathrm{x}-1)(\mathrm{x}+1)}$