What is the value of the sum \sum_{n=1}^{20}(i^{n-1}+i^{n}+i^{n+1}) where i=\sqrt{-1}?
- A. -2i
- B. 0 ✓
- C. 1
- D. 2i
Correct Answer: B. 0
Explanation
The terms inside the sum simplify to i^{n-1}(1 + i + i^2) = i^{n-1}(1 + i - 1) = i^n. The summation is \sum_{n=1}^{20} i^n. Since the sum of any four consecutive powers of i is zero and 20 is a multiple of 4, the total sum is 0.
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