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GATE Overflow Test Series | Quantitative Aptitude | Test 1 | Question: 14

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The value of $i^k + i^{k + 1} + i^{k + 2} + i^{k + 3} = 0,$ where $k$ is an integer.

"The sum of any four consecutive powers of $i$ is equal to $0.$"

i.e., $i + i^{2} + i^{3} + i^{4} = 0$

We can write the given expression as

$(i^{1} + i^{2} + i^{3} + i^{4}) + (i^{5} + i^{6} + i^{7} + i^{8}) + \dots  + (i^{213} + i^{2014} + i^{2015} + i^{2016}) + (i^{2017} + i^{2018} + i^{2019} + i^{2020}) + i^{2021}$

Now, from the above result, the value in every bracket becomes $0$ as the sum of $4$ consecutive powers of $i$ is equal to $0.$ There are $505$ brackets and every bracket equals to $0.$

Thus, $505(0) + i^{2021} = 0 + i^{1} = i.$

So, the correct answer is $(B).$
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