`i^(14)+i^(15)+i^(16)+i^(17)=`

Options

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

Explanation

To solve the expression \( i^{14} + i^{15} + i^{16} + i^{17} \), we will follow these steps:

### Step 1: Understand the powers of \( i \)
The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). The powers of \( i \) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \), and so on.

### Step 2: Calculate each power in the expression
Now, we will calculate each term in the expression:

1. **Calculate \( i^{14} \)**:
\[ 14 \mod 4 = 2 \quad \Rightarrow \quad i^{14} = i^2 = -1 \]

2. **Calculate \( i^{15} \)**:
\[ 15 \mod 4 = 3 \quad \Rightarrow \quad i^{15} = i^3 = -i \]

3. **Calculate \( i^{16} \)**:
\[ 16 \mod 4 = 0 \quad \Rightarrow \quad i^{16} = i^0 = 1 \]

4. **Calculate \( i^{17} \)**:
\[ 17 \mod 4 = 1 \quad \Rightarrow \quad i^{17} = i^1 = i \]

### Step 3: Substitute the values back into the expression
Now we can substitute the calculated values back into the original expression:
\[ i^{14} + i^{15} + i^{16} + i^{17} = -1 + (-i) + 1 + i \]

### Step 4: Simplify the expression
Combine the terms:
\[ (-1 + 1) + (-i + i) = 0 + 0 = 0 \]

### Final Answer
Thus, the value of \( i^{14} + i^{15} + i^{16} + i^{17} \) is:
\[ \boxed{0} \]