If i`=sqrt1` and n is a positive integer, which of the following statements is FALSE?

Options

A. `i^(4n)=1`
B. `i^(4n+1)=-i`
C. `i^(4n+2)=-1`
D. `i^(n+4)=i^n`

Explanation

To solve the problem, we need to analyze the properties of the imaginary unit \( i \), where \( i = \sqrt{-1} \), and evaluate the given statements to determine which one is false.

### Step-by-Step Solution:

1. **Understanding the properties of \( i \)**:
- We know that:
- \( i^2 = -1 \)
- \( i^3 = i^2 \cdot i = -1 \cdot i = -i \)
- \( i^4 = (i^2)^2 = (-1)^2 = 1 \)

2. **Evaluating each statement**:
- **Option A**: \( i^{4n} = 1 \)
- Since \( i^4 = 1 \), we can write:
\[ i^{4n} = (i^4)^n = 1^n = 1 \]
- This statement is **true**.

- **Option B**: \( i^{4n+1} = -i \)
- We can express \( i^{4n+1} \) as:
\[ i^{4n+1} = i^{4n} \cdot i = 1 \cdot i = i \]
- This means \( i^{4n+1} \neq -i \). Therefore, this statement is **false**.

- **Option C**: \( i^{4n+2} = -1 \)
- We can express \( i^{4n+2} \) as:
\[ i^{4n+2} = i^{4n} \cdot i^2 = 1 \cdot (-1) = -1 \]
- This statement is **true**.

- **Option D**: \( i^{n+4} = i^n \)
- We can express \( i^{n+4} \) as:
\[ i^{n+4} = i^n \cdot i^4 = i^n \cdot 1 = i^n \]
- This statement is **true**.

3. **Conclusion**:
- The only statement that is false is **Option B**: \( i^{4n+1} = -i \).

### Final Answer:
The false statement is **Option B**: \( i^{4n+1} = -i \).