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

Question

Write the sum of the series $\text{i}+\text{i}^2+\text{i}^3+...$ upto 1000 terms.

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Answer

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

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