If `x_(1),x_(2),x_(3),x_(4),x_(5)` are five consecutive odd numbers then their average is

Options

A. `x_(2)`
B. `x_(3)`
C. `x_(4)`
D. `x_(5)`

Answer

To find the average of five consecutive odd numbers \(x_1, x_2, x_3, x_4, x_5\), we can follow these steps:

### Step-by-Step Solution:

1. **Define the First Odd Number**:
Let the first odd number be \(x_1\).

2. **Express the Consecutive Odd Numbers**:
Since they are consecutive odd numbers, we can express them as:
- \(x_2 = x_1 + 2\)
- \(x_3 = x_1 + 4\)
- \(x_4 = x_1 + 6\)
- \(x_5 = x_1 + 8\)

3. **Sum the Five Odd Numbers**:
Now, we need to find the sum of these five numbers:
\[ \text{Sum} = x_1 + x_2 + x_3 + x_4 + x_5 \]
Substituting the expressions for \(x_2, x_3, x_4, x_5\):
\[ \text{Sum} = x_1 + (x_1 + 2) + (x_1 + 4) + (x_1 + 6) + (x_1 + 8) \]
Simplifying this:
\[ \text{Sum} = x_1 + x_1 + 2 + x_1 + 4 + x_1 + 6 + x_1 + 8 \]
\[ \text{Sum} = 5x_1 + (2 + 4 + 6 + 8) \]
\[ \text{Sum} = 5x_1 + 20 \]

4. **Calculate the Average**:
The average is calculated by dividing the sum by the number of observations (which is 5):
\[ \text{Average} = \frac{\text{Sum}}{5} = \frac{5x_1 + 20}{5} \]
Simplifying this:
\[ \text{Average} = x_1 + 4 \]

5. **Identify the Average in Terms of Consecutive Odd Numbers**:
Notice that \(x_1 + 4\) is actually the third odd number \(x_3\) in our sequence of consecutive odd numbers. Thus, we can conclude:
\[ \text{Average} = x_3 \]

### Final Answer:
The average of the five consecutive odd numbers \(x_1, x_2, x_3, x_4, x_5\) is \(x_3\).

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