Question: Which of the following is equivalent to $i^{50}$?
$-i$
$i$
$-1$
$1$
1. **State the problem:** We need to find which of the given options is equivalent to $i^{50}$, where $i$ is the imaginary unit with the property $i^2 = -1$.
2. **Recall the powers of $i$:** The powers of $i$ cycle every 4 steps:
$$i^1 = i$$
$$i^2 = -1$$
$$i^3 = -i$$
$$i^4 = 1$$
Then the cycle repeats: $i^5 = i$, $i^6 = -1$, and so on.
3. **Find the remainder when 50 is divided by 4:**
$$50 \div 4 = 12 \text{ remainder } 2$$
So,
$$50 = 4 \times 12 + 2$$
4. **Use the cyclic property:**
$$i^{50} = i^{4 \times 12 + 2} = (i^4)^{12} \times i^2 = 1^{12} \times (-1) = -1$$
5. **Conclusion:**
The value of $i^{50}$ is $-1$.
**Final answer:** $-1$