Question:
Find the sum of the GP :
$1-a+a^{2}-a^{3}+\ldots$ to $n$ terms $(a \neq 1)$
Solution:
Sum of a G.P. series is represented by the formula $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac{\mathrm{r}^{\mathrm{n}}-1}{\mathrm{r}-1}$
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 = 1
$r=$ (ratio between the $n$ term and $n-1$ term) $-a \div 1=-a$
n terms
$\therefore \mathrm{S}_{\mathrm{n}}=1 \times \frac{(-\mathrm{a})^{\mathrm{n}}-1}{-\mathrm{a}-1}$
[Multiplying both numerator and denominator by -1]
$\Rightarrow \mathrm{S}_{\mathrm{n}}=\frac{1-(-\mathrm{a})^{\mathrm{n}}}{1+\mathrm{a}}$