Question

The value of the sum \( \sum_{n=1}^{13} (i^n + i^{n+1}) \), where \( i = \sqrt{-1} \), equals:

• A. \( i \)
• B. \( i - 1 \)
• C. \( -i \)
• D. \( 0 \)

Hint:

The powers of \( i \) repeat every four terms, so sums involving \( i^n \) can be simplified by recognizing the periodicity.

Answer 1

The Correct Option is D

Step 1: Sum of powers of \( i \). The powers of \( i \) repeat every 4 terms: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Thus, the sum \( \sum_{n=1}^{13} (i^n + i^{n+1}) \) simplifies to 0 after applying this periodicity.
Step 2: Conclusion. Thus, the sum is equal to 0.