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Question
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
Options
1
−1
i
0
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Solution
0
\[i + i^2 + i^3 + i^4 . . . i^{1000} \]
\[ i + i^2 + i^3 + i^4 [ \because i^2 = - 1, i^3 = - i \text { and } i^4 = 1]\]
\[ = i - 1 - i + 1 \]
\[ = 0 \]
\[\text { Similarly, the sum of the next four terms of the series will be equal to 0 . This is because the powers of i follow a cyclicity of 4 } . \]
\[\text { Hence, the sum of all terms, till 1000, will be zero } . \]
\[i + i^2 + i^3 + i^4 . . . i^{1000} = 0\]
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