If m,n,p,q are consecutive integers then the value of `i^(m)+i^(n)+i^(p)+i^(q)` is

Options

A. 1
B. 4
C. 0
D. none of these

Explanation

To solve the problem, we need to evaluate the expression \( i^m + i^n + i^p + i^q \) where \( m, n, p, q \) are consecutive integers.

### Step-by-step Solution:

1. **Define the Consecutive Integers**:
Let \( m \) be the first integer. Then the consecutive integers can be defined as:
- \( n = m + 1 \)
- \( p = m + 2 \)
- \( q = m + 3 \)

2. **Substitute the Values**:
Now substitute these values into the expression:
\[ i^m + i^{m+1} + i^{m+2} + i^{m+3} \]

3. **Factor out \( i^m \)**:
We can factor out \( i^m \) from the expression:
\[ i^m (1 + i^1 + i^2 + i^3) \]

4. **Evaluate \( 1 + i + i^2 + i^3 \)**:
We know the powers of \( i \):
- \( i^0 = 1 \)
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)

Therefore, we can substitute these values:
\[ 1 + i + (-1) + (-i) = 0 \]

5. **Final Expression**:
Now substituting back, we have:
\[ i^m \cdot 0 = 0 \]

Thus, the value of \( i^m + i^n + i^p + i^q \) is \( \boxed{0} \).