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Question
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Solution
\[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3} \]
\[ = i^n + i^n . i + i^n . i^2 + i^n . i^3 \]
\[ = i^n + i^n . i + i^n . ( - 1) + i^n . ( - i)\]
\[ = i^n + i^n . i - i^n - i^n . i\]
\[ = 0\]
Thus, the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] is 0.
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