Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers1 Mark
Question
Write the sum of the series $\text{i}+\text{i}^2+\text{i}^3+...$ upto 1000 terms.
✓
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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