Complex Numbers

Question

If i = √(-1), then i⁴ + i⁸ + i¹₂ + i¹⁶ equals

RankGuru · Accepted Answer

1. **Powers of $i$ Cycle**: The powers of $i$ repeat every four terms:
   - $i^1 = i$
   - $i^2 = -1$
   - $i^3 = -i$
   - $i^4 = 1$
   - In general, $i^{4k} = 1$ for any integer $k$.
2. **Apply to Question**: 
   - $i^4 = 1$
   - $i^8 = 1$ ($8$ is a multiple of $4$)
   - $i^{12} = 1$ ($12$ is a multiple of $4$)
   - $i^{16} = 1$ ($16$ is a multiple of $4$)
3. **Summation**: $1 + 1 + 1 + 1 = 4$.

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