Question:
Which ONE of the following is the value of \( \frac{1}{i^n} \)?
where \( i = \sqrt{-1} \), and \( n \) is an even positive integer.
Which ONE of the following is the value of \( \frac{1}{i^n} \)?
where \( i = \sqrt{-1} \), and \( n \) is an even positive integer.
where \( i = \sqrt{-1} \), and \( n \) is an even positive integer.
Show Hint
Powers of i repeat every 4 steps: i, -1, -i, 1. For even n, i^n is always +1 or -1.
Updated On: Aug 5, 2026
- +1 or \( -1 \)
- \( +i \) or \( -i \)
- only +1
- only \( -i \)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We need to find the value of \( \frac{1}{i^n} \) where \( i = \sqrt{-1} \) and \( n \) is an even positive integer.
The answer needs to hold for any even n, not just one specific value.
Step 2: Key Formula or Approach:
The powers of \( i \) repeat in a cycle of 4: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \).
Since n is even, write \( n = 2k \) for some positive integer k, so \( i^n = (i^2)^k = (-1)^k \).
Step 3: Detailed Explanation:
If k is even, \( (-1)^k = 1 \), so \( i^n = 1 \) and \( \frac{1}{i^n} = 1 \).
If k is odd, \( (-1)^k = -1 \), so \( i^n = -1 \) and \( \frac{1}{i^n} = -1 \).
So depending on whether n is a multiple of 4 or just an even number that is not a multiple of 4, the value swings between +1 and -1.
Check option (B): +i or -i only happens for odd powers of i, not even ones, so this is wrong.
Check option (C): only +1 is true when n is a multiple of 4 but fails for n = 2, 6, 10, so this is incomplete.
Check option (D): only -i never happens for even powers at all, so this is wrong.
\[ \frac{1}{i^n} = \frac{1}{(-1)^k} = \pm 1 \]
Final Answer:
The value of \( \frac{1}{i^n} \) is +1 or -1 depending on n, so option (A) is correct. \[ \boxed{+1 \text{ or } -1} \]
We need to find the value of \( \frac{1}{i^n} \) where \( i = \sqrt{-1} \) and \( n \) is an even positive integer.
The answer needs to hold for any even n, not just one specific value.
Step 2: Key Formula or Approach:
The powers of \( i \) repeat in a cycle of 4: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \).
Since n is even, write \( n = 2k \) for some positive integer k, so \( i^n = (i^2)^k = (-1)^k \).
Step 3: Detailed Explanation:
If k is even, \( (-1)^k = 1 \), so \( i^n = 1 \) and \( \frac{1}{i^n} = 1 \).
If k is odd, \( (-1)^k = -1 \), so \( i^n = -1 \) and \( \frac{1}{i^n} = -1 \).
So depending on whether n is a multiple of 4 or just an even number that is not a multiple of 4, the value swings between +1 and -1.
Check option (B): +i or -i only happens for odd powers of i, not even ones, so this is wrong.
Check option (C): only +1 is true when n is a multiple of 4 but fails for n = 2, 6, 10, so this is incomplete.
Check option (D): only -i never happens for even powers at all, so this is wrong.
\[ \frac{1}{i^n} = \frac{1}{(-1)^k} = \pm 1 \]
Final Answer:
The value of \( \frac{1}{i^n} \) is +1 or -1 depending on n, so option (A) is correct. \[ \boxed{+1 \text{ or } -1} \]
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