Question

Evaluate the following sum: \[ \sum_{n=1}^{2025} i^n (1+i) \]

• A. \( 2025i \)
• B. \( 2025(1+i) \)
• C. \( 2025 \)
• D. None of the above

Hint:

When summing powers of \( i \), recognize the repeating pattern every 4 terms to simplify the sum.

Answer 1

The Correct Option is B

We are asked to evaluate the sum: \[ S = \sum_{n=1}^{2025} i^n (1+i) \] The powers of \( i \) cycle every four terms: \[ i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and then it repeats.} \] Thus, the sum can be written as: \[ S = \sum_{n=1}^{2025} i^n (1+i) \] Since the powers of \( i \) repeat every 4 terms, we can calculate the sum of one complete cycle: \[ S_{\text{cycle}} = i(1+i) + (-1)(1+i) + (-i)(1+i) + 1(1+i) \] Simplifying each term: \[ S_{\text{cycle}} = i + i^2 + (-i) + i + 1 + i = 2i + 1 \] Since there are 2025 terms, and 2025 is divisible by 4 (2025 = 4 * 506 + 1), we can split the sum into complete cycles plus the last term: \[ S = 506 \times (2i + 1) + (i(1+i)) = 506 \times (2i + 1) + i + i^2 = 506 \times (2i + 1) + i - 1 \] Finally, the sum simplifies to: \[ S = 2025(1+i) \] Thus, the correct answer is: \[ \boxed{(B) 2025(1+i)} \]