Question

Write the sum of the series $i+i^2+i^3+....$upto 1000 terms.

Answer 1

We know that,
$i+i^2+i^3+i^4=i-1-i+1=0$

$$\begin{gathered} \therefore i+i^2+i^3+....+i^{1000} \\ =\left(i+i^2+i^3+i^4\right)+\left(i^5+i^6+i^7+i^8\right)+...+\left(i^{997}+i^{998}+i^{999}+i^{1000}\right) \\ =\left(i+i^2+i^3+i^4\right)+\left(i^{4}i+i^4{i}^2+i^4{i}^3+i^4{i}^4\right)+...+\left[{\left(i^4\right)}^{249}i+{\left(i^4\right)}^{249}{i}^2+{\left(i^4\right)}^{249}{i}^3+{\left(i^4\right)}^{249}{i}^4\right] \\ =\left(i+i^2+i^3+i^4\right)+\left(i+i^2+i^3+i^4\right)+...+\left(i+i^2+i^3+i^4\right) \\ =0 \end{gathered}$$

Thus, the sum of the series $i+i^2+i^3+....$upto 1000 terms is 0.