`a_1,a_2, a_3, ,a_n` are in A.P. and `a_1+a_3+a_5+a_7+a_9=20` then `a_5` is `2` 2. `7` 3. `3` 4. `4` 5. 5

Options

A. `2`
B. `7`
C. `3`
D. `4`

Explanation

To solve the problem, we need to find the value of \( a_5 \) given that \( a_1, a_2, a_3, \ldots, a_n \) are in Arithmetic Progression (A.P.) and that \( a_1 + a_3 + a_5 + a_7 + a_9 = 20 \).

### Step-by-Step Solution:

1. **Identify the terms in A.P.**:
In an A.P., the \( n \)-th term can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \( d \) is the common difference.

2. **Express the given terms**:
We can express the terms \( a_1, a_3, a_5, a_7, a_9 \) in terms of \( a_1 \) and \( d \):
- \( a_1 = a_1 \)
- \( a_3 = a_1 + 2d \)
- \( a_5 = a_1 + 4d \)
- \( a_7 = a_1 + 6d \)
- \( a_9 = a_1 + 8d \)

3. **Set up the equation**:
Now, substituting these expressions into the equation:
\[ a_1 + (a_1 + 2d) + (a_1 + 4d) + (a_1 + 6d) + (a_1 + 8d) = 20 \]

4. **Combine like terms**:
Combine the terms:
\[ 5a_1 + (2d + 4d + 6d + 8d) = 20 \]
This simplifies to:
\[ 5a_1 + 20d = 20 \]

5. **Simplify the equation**:
Divide the entire equation by 5:
\[ a_1 + 4d = 4 \]

6. **Identify \( a_5 \)**:
Recall that \( a_5 = a_1 + 4d \). From the equation we derived:
\[ a_5 = 4 \]

### Conclusion:
Thus, the value of \( a_5 \) is \( 4 \).