The value of `i^101 + i^102 + i^103 + .....+i^108` is :

Options

A. 8
B. -1
C. 0
D. -2

Answer

To solve the expression \( i^{101} + i^{102} + i^{103} + \ldots + i^{108} \), we first need to understand the powers of \( i \), where \( i \) is the imaginary unit defined as \( i = \sqrt{-1} \).

### Step-by-Step Solution:

1. **Identify the Pattern of Powers of \( i \)**:
The powers of \( i \) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and the cycle repeats)

2. **Find the Remainders**:
To simplify \( i^{101}, i^{102}, i^{103}, \ldots, i^{108} \), we can find the remainders when these exponents are divided by 4:
- \( 101 \mod 4 = 1 \) → \( i^{101} = i^1 = i \)
- \( 102 \mod 4 = 2 \) → \( i^{102} = i^2 = -1 \)
- \( 103 \mod 4 = 3 \) → \( i^{103} = i^3 = -i \)
- \( 104 \mod 4 = 0 \) → \( i^{104} = i^0 = 1 \)
- \( 105 \mod 4 = 1 \) → \( i^{105} = i^1 = i \)
- \( 106 \mod 4 = 2 \) → \( i^{106} = i^2 = -1 \)
- \( 107 \mod 4 = 3 \) → \( i^{107} = i^3 = -i \)
- \( 108 \mod 4 = 0 \) → \( i^{108} = i^0 = 1 \)

3. **Substitute the Values**:
Now substitute these values back into the original expression:
\[ i^{101} + i^{102} + i^{103} + i^{104} + i^{105} + i^{106} + i^{107} + i^{108} = i + (-1) + (-i) + 1 + i + (-1) + (-i) + 1 \]

4. **Combine Like Terms**:
Now, combine the terms:
- Combine the \( i \) terms: \( i - i + i - i = 0 \)
- Combine the real terms: \( -1 + 1 - 1 + 1 = 0 \)

5. **Final Result**:
Therefore, the sum is:
\[ 0 + 0 = 0 \]

### Final Answer:
The value of \( i^{101} + i^{102} + i^{103} + \ldots + i^{108} \) is \( \boxed{0} \).