Series

We are asked to compute the sum of the series: This can be rewritten as: Thus, the series becomes a telescoping series: All intermediate terms cancel out, and we are left with: Thus, the sum is: Thus, the correct answer is (C).

We are asked to evaluate the sum: The powers of cycle every four terms: Thus, the sum can be written as: Since the powers of repeat every 4 terms, we can calculate the sum of one complete cycle: Simplifying each term: 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: Finally, the sum simplifies to: Thus, the correct answer is: