The value of `i^(2n) + i^(2n+1) + i^(2n+2) + i^(2n+3)`, where `i = sqrt( -1)` is `:`

Options

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

Answer

To solve the expression \( i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3} \), where \( i = \sqrt{-1} \), we can follow these steps:

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

### Step 2: Simplify each term in the expression
We can express each term in the given expression based on the value of \( n \):

1. \( i^{2n} \): Since \( 2n \) is even, \( i^{2n} = (-1)^n \).
2. \( i^{2n+1} \): This is \( i^{2n} \cdot i = (-1)^n \cdot i \).
3. \( i^{2n+2} \): This is \( i^{2n} \cdot i^2 = (-1)^n \cdot (-1) = -(-1)^n \).
4. \( i^{2n+3} \): This is \( i^{2n} \cdot i^3 = (-1)^n \cdot (-i) = -(-1)^n \cdot i \).

### Step 3: Combine the terms
Now, we can combine all these terms:
\[ i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3} = (-1)^n + (-1)^n i - (-1)^n - (-1)^n i \]

### Step 4: Simplify the expression
Notice that \( (-1)^n \) and \( -(-1)^n \) cancel each other out:
\[ (-1)^n + (-1)^n i - (-1)^n - (-1)^n i = 0 \]

### Conclusion
Thus, the value of \( i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3} \) is:
\[ \boxed{0} \]