In n is an even integer, which one of the following is an odd integer?

Options

A. `n^2`
B. `(n+1)//2`
C. `-2n - 4`
D. `2n^2 - 3`

Answer

To solve the problem of finding which expression yields an odd integer when \( n \) is an even integer, we will evaluate each option step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that \( n \) is an even integer. An even integer can be expressed in the form \( n = 2k \), where \( k \) is an integer.

2. **Assuming a Value for \( n \)**:
To simplify our calculations, let's assume \( n = 2 \) (the smallest even integer).

3. **Evaluating Each Option**:
We will evaluate each option one by one to see which one gives an odd integer.

- **Option A**: \( n^2 \)
\[ n^2 = 2^2 = 4 \]
Since 4 is even, option A is not the answer.

- **Option B**: \( \frac{n + 1}{2} \)
\[ \frac{n + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2} = 1.5 \]
Since 1.5 is not an integer, option B is not the answer.

- **Option C**: \( -2n - 4 \)
\[ -2n - 4 = -2(2) - 4 = -4 - 4 = -8 \]
Since -8 is even, option C is not the answer.

- **Option D**: \( 2n^2 - 3 \)
\[ 2n^2 - 3 = 2(2^2) - 3 = 2(4) - 3 = 8 - 3 = 5 \]
Since 5 is odd, option D is the answer.

4. **Conclusion**:
The expression that yields an odd integer when \( n \) is an even integer is:
\[ \text{Option D: } 2n^2 - 3 \]