2n is an even number. What are the odd numbers on each side of it ? The sum of two consecutive odd numbers is 96. What are they ?

Options

A. 41, 43
B. 57, 59
C. 47, 49
D. 31, 33

Answer

To solve the problem step by step, we will break it down into two parts: finding the odd numbers on each side of the even number \(2n\) and then finding the two consecutive odd numbers whose sum is 96.

### Part 1: Finding Odd Numbers on Each Side of \(2n\)

1. **Identify the even number**: We are given that \(2n\) is an even number.

2. **Determine the odd numbers**:
- The odd number just before \(2n\) can be found by subtracting 1 from \(2n\):
\[ \text{Odd number before } 2n = 2n - 1 \]
- The odd number just after \(2n\) can be found by adding 1 to \(2n\):
\[ \text{Odd number after } 2n = 2n + 1 \]

3. **Conclusion**: The odd numbers on each side of \(2n\) are:
\[ 2n - 1 \quad \text{and} \quad 2n + 1 \]

### Part 2: Finding Two Consecutive Odd Numbers that Sum to 96

1. **Define the consecutive odd numbers**: Let the first odd number be \(x\). The next consecutive odd number can be expressed as \(x + 2\).

2. **Set up the equation**: According to the problem, the sum of these two consecutive odd numbers is 96:
\[ x + (x + 2) = 96 \]

3. **Simplify the equation**:
\[ 2x + 2 = 96 \]

4. **Isolate \(x\)**:
- Subtract 2 from both sides:
\[ 2x = 94 \]
- Divide by 2:
\[ x = 47 \]

5. **Find the two consecutive odd numbers**:
- The first odd number is \(x = 47\).
- The second odd number is \(x + 2 = 49\).

6. **Conclusion**: The two consecutive odd numbers are:
\[ 47 \quad \text{and} \quad 49 \]

### Final Answer
- The odd numbers on each side of \(2n\) are \(2n - 1\) and \(2n + 1\).
- The two consecutive odd numbers whose sum is 96 are \(47\) and \(49\).