Question

Evaluate \( i^2 + i^3 + \dots + i^{4000} \):

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

Hint:

When working with powers of \( i \), recognize the repeating cycle \( (i, -1, -i, 1) \) and simplify by grouping terms accordingly.

Answer 1

The Correct Option is D

The powers of \( i \) follow a repeating cycle every 4 terms: \[ i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and then the cycle repeats}. \] We can express the sum \( i^2 + i^3 + \dots + i^{4000} \) as a series of repeated cycles: \[ (i^2 + i^3 + i^4 + i^1) + (i^2 + i^3 + i^4 + i^1) + \dots \text{(repeated 1000 times)}. \] Each cycle simplifies to: \[ i^2 + i^3 + i^4 + i^1 = -1 - i + 1 + i = 0. \] Since each group sums to zero, the overall sum of the cycles is: \[ 0 \cdot 1000 = 0. \] Now, consider the leftover terms, \( i^2 \) and \( i^3 \): \[ i^2 + i^3 = -1 + i. \] Thus, the final sum is: \[ \boxed{-i}. \]