Question

The value of $ i^3 + i^4 + i^5 $+ .. + $i^{93}$ is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( -1 \)
• D. \( 2 \)

Hint:

The powers of \( i \) follow a cycle of 4 terms, so break the sum into groups of four and simplify accordingly.

Answer 1

The Correct Option is A

We are asked to find: \[ i^3 + i^4 + i^5 + \ldots + i^{93} \] We know that the powers of \( i \) (the imaginary unit) follow a cyclic pattern: \[ i^1 = i, \, i^2 = -1, \, i^3 = -i, \, i^4 = 1, \, i^5 = i, \, \ldots \] The powers of \( i \) repeat every 4 terms. Since \( 93 \div 4 = 23 \) remainder 1, we have 23 complete cycles of 4 terms, plus 1 extra term, which is \( i^3 \).
Thus, we have: \[ 23 \times (i^1 + i^2 + i^3 + i^4) + i^3 \] The sum of one complete cycle is: \[ i^1 + i^2 + i^3 + i^4 = i + (-1) + (-i) + 1 = 0 \]
Thus, the entire sum is: \[ 23 \times 0 + i^3 = 0 + (-i) = -i \] Therefore, the correct answer is \( 0 \).