For positive whole number of n, what is the value of `i^(4n+1)` ?

Options

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

Answer

To find the value of \( i^{4n+1} \) for a positive whole number \( n \), we can 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:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

### Step 2: Break down the exponent \( 4n + 1 \)
We can express \( i^{4n+1} \) as:
\[ i^{4n+1} = i^{4n} \cdot i^1 \]

### Step 3: Simplify \( i^{4n} \)
Since \( i^4 = 1 \), we can use this to simplify \( i^{4n} \):
\[ i^{4n} = (i^4)^n = 1^n = 1 \]

### Step 4: Combine the results
Now substituting back into our expression:
\[ i^{4n+1} = i^{4n} \cdot i^1 = 1 \cdot i = i \]

### Conclusion
Thus, the value of \( i^{4n+1} \) is:
\[ \boxed{i} \]