Question:
Write the sum of the series $i+i^{2}+i^{3}+\ldots$ upto 1000 terms.
Solution:
We know that,
$i+i^{2}+i^{3}+i^{4}=i-1-i+1=0$
$\therefore i+i^{2}+i^{3}+\ldots+i^{1000}$
$=\left(i+i^{2}+i^{3}+i^{4}\right)+\left(i^{5}+i^{6}+i^{7}+i^{8}\right)+\ldots+\left(i^{997}+i^{998}+i^{999}+i^{1000}\right)$
$=\left(i+i^{2}+i^{3}+i^{4}\right)+\left(i^{4} i+i^{4} i^{2}+i^{4} i^{3}+i^{4} i^{4}\right)+\ldots+\left[\left(i^{4}\right)^{249} i+\left(i^{4}\right)^{249} i^{2}+\left(i^{4}\right)^{249} i^{3}+\left(i^{4}\right)^{249} i^{4}\right]$
$=\left(i+i^{2}+i^{3}+i^{4}\right)+\left(i+i^{2}+i^{3}+i^{4}\right)+\ldots+\left(i+i^{2}+i^{3}+i^{4}\right)$
$=0$
Thus, the sum of the series $i+i^{2}+i^{3}+\ldots$ upto 1000 terms is 0 .