Question

Two numbers \(k_1\) and \(k_2\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \(i^{k_1} + i^{k_2}\) (where \(i = \sqrt{-1}\)) is non-zero equals:

• A. \(\frac{3}{4}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{2}{3}\)
• D. \(\frac{1}{4} \)

Hint:

Understanding the cyclic properties of complex numbers can significantly simplify probability calculations involving their powers.

Answer 1

The Correct Option is A

Step 1: Understand the cyclic nature of \(i\).
Note that \(i\) cycles every four powers: \(i, -1, -i, 1\). Thus, \(i^k\) repeats every four values.
Step 2: Calculate zero-sum pairs.
Identify pairs \((k_1, k_2)\) such that \(i^{k_1} + i^{k_2} = 0\).
These occur when \(k_1\) and \(k_2\) are opposites in the cycle (e.g., \(i\) and \(-i\), \(1\) and \(-1\)).
Step 3: Calculate probabilities.
The probability of a zero-sum pair given the four-cycle nature is \( \frac{1}{4} \) (since each opposite appears once in the cycle).
Step 4: Determine the non-zero probability.
Since the probability of a zero-sum is \( \frac{1}{4} \), the probability of a non-zero sum is \( 1 - \frac{1}{4} = \frac{3}{4} \).
Conclusion: The probability that the sum \(i^{k_1} + i^{k_2}\) is non-zero is \( \frac{3}{4} \).