Question:
Given that $i^2 = -1$. Then $i^{13} + i^{14} + i^{15} + \ldots + i^{2026}$ is equal to:
Given that $i^2 = -1$. Then $i^{13} + i^{14} + i^{15} + \ldots + i^{2026}$ is equal to:
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Powers of $i$ repeat every 4 terms. Always reduce exponent modulo 4.
Updated On: Apr 24, 2026
- $i - 2$
- $i + 2$
- $2i + 1$
- $i - 1$
- $-i + 1$
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The Correct Option is D
Solution and Explanation
Concept:
• Powers of $i$ are cyclic with period 4: \[ i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1 \]
• This cycle repeats for all higher powers
Step 1: Identify repeating pattern
Each group of 4 terms sums to: \[ i + (-1) + (-i) + 1 = 0 \]
Step 2: Count number of terms
From $13$ to $2026$: \[ \text{Number of terms} = 2026 - 13 + 1 = 2014 \]
Step 3: Divide into complete cycles
\[ 2014 \div 4 = 503 \text{ cycles with remainder } 2 \]
Step 4: Sum remaining terms
Remaining terms: \[ i^{13}, i^{14} \] \[ i^{13} = i,\quad i^{14} = -1 \] \[ \text{Sum} = i - 1 \] Final Conclusion:
\[ i^{13} + i^{14} + \cdots + i^{2026} = i - 1 \]
• Powers of $i$ are cyclic with period 4: \[ i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1 \]
• This cycle repeats for all higher powers
Step 1: Identify repeating pattern
Each group of 4 terms sums to: \[ i + (-1) + (-i) + 1 = 0 \]
Step 2: Count number of terms
From $13$ to $2026$: \[ \text{Number of terms} = 2026 - 13 + 1 = 2014 \]
Step 3: Divide into complete cycles
\[ 2014 \div 4 = 503 \text{ cycles with remainder } 2 \]
Step 4: Sum remaining terms
Remaining terms: \[ i^{13}, i^{14} \] \[ i^{13} = i,\quad i^{14} = -1 \] \[ \text{Sum} = i - 1 \] Final Conclusion:
\[ i^{13} + i^{14} + \cdots + i^{2026} = i - 1 \]
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