Question:
The sum of $i^2 + i^4 + \cdots$ upto 25 terms is equal to
The sum of $i^2 + i^4 + \cdots$ upto 25 terms is equal to
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Use cyclic nature of powers of $i$ (cycle length = 4).
Updated On: Apr 30, 2026
- 0
- $i$
- $-i$
- 1
- $-1$
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The Correct Option is
Solution and Explanation
Concept:
Powers of $i$ repeat in cycle of 4.
\[
i^1=i,\; i^2=-1,\; i^3=-i,\; i^4=1
\]
Step 1: Observe pattern
\[ i^2 = -1,\quad i^4 = 1,\quad i^6 = -1,\quad i^8 = 1 \] So terms alternate: $-1, 1, -1, 1, \dots$
Step 2: Group pairs
Each pair: \[ -1 + 1 = 0 \]
Step 3: Total terms
25 terms → 12 pairs + 1 extra term Extra term = $i^{50}$ \[ i^{50} = i^{2} = -1 \] Final Conclusion:
Sum = $-1$
Step 1: Observe pattern
\[ i^2 = -1,\quad i^4 = 1,\quad i^6 = -1,\quad i^8 = 1 \] So terms alternate: $-1, 1, -1, 1, \dots$
Step 2: Group pairs
Each pair: \[ -1 + 1 = 0 \]
Step 3: Total terms
25 terms → 12 pairs + 1 extra term Extra term = $i^{50}$ \[ i^{50} = i^{2} = -1 \] Final Conclusion:
Sum = $-1$
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