What is `i^(100)+i^(1001)+i^(1002)+i^(1003)` equal to (where `i=sqrt(-1))` ?

Options

A. 0
B. `i`
C. `-i`
D. 1

Explanation

To solve the expression \( i^{100} + i^{1001} + i^{1002} + i^{1003} \), where \( i = \sqrt{-1} \), we can follow these steps:

### Step 1: Identify the pattern in powers of \( i \)
The powers of \( i \) cycle every 4:
- \( i^0 = 1 \)
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \) (and the cycle repeats)

### Step 2: Reduce the exponents modulo 4
To find \( i^{100} \), \( i^{1001} \), \( i^{1002} \), and \( i^{1003} \), we can reduce each exponent modulo 4.

- \( 100 \mod 4 = 0 \) → \( i^{100} = i^0 = 1 \)
- \( 1001 \mod 4 = 1 \) → \( i^{1001} = i^1 = i \)
- \( 1002 \mod 4 = 2 \) → \( i^{1002} = i^2 = -1 \)
- \( 1003 \mod 4 = 3 \) → \( i^{1003} = i^3 = -i \)

### Step 3: Substitute the values back into the expression
Now we can substitute these values back into the original expression:
\[ i^{100} + i^{1001} + i^{1002} + i^{1003} = 1 + i - 1 - i \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ 1 - 1 + i - i = 0 + 0 = 0 \]

### Final Answer
Thus, the value of \( i^{100} + i^{1001} + i^{1002} + i^{1003} \) is \( \boxed{0} \).

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