If `a_(1),a_(2)a_(3),….,a_(15)` are in `A.P` and `a_(1)+a_(8)+a_(15)=15`, then `a_(2)+a_(3)+a_(8)+a_(13)+a_(14)` is equal to

Options

A. `25`
B. `35`
C. `10`
D. `15`

Explanation

To solve the problem, we need to find the value of \( a_2 + a_3 + a_8 + a_{13} + a_{14} \) given that \( a_1 + a_8 + a_{15} = 15 \) and the terms \( a_1, a_2, \ldots, a_{15} \) are in an arithmetic progression (A.P.).

### Step-by-Step Solution:

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

2. **Finding \( a_8 \)**:
The 8th term can be expressed as:
\[ a_8 = a_1 + 7d \]
The 15th term can be expressed as:
\[ a_{15} = a_1 + 14d \]

3. **Using the Given Condition**:
We know:
\[ a_1 + a_8 + a_{15} = 15 \]
Substituting \( a_8 \) and \( a_{15} \):
\[ a_1 + (a_1 + 7d) + (a_1 + 14d) = 15 \]
Simplifying this gives:
\[ 3a_1 + 21d = 15 \]
Dividing the entire equation by 3:
\[ a_1 + 7d = 5 \quad \text{(Equation 1)} \]

4. **Finding \( a_2, a_3, a_{13}, a_{14} \)**:
Now, we can express \( a_2, a_3, a_{13}, a_{14} \):
\[ a_2 = a_1 + d \]
\[ a_3 = a_1 + 2d \]
\[ a_{13} = a_1 + 12d \]
\[ a_{14} = a_1 + 13d \]

5. **Calculating the Required Sum**:
We need to find:
\[ a_2 + a_3 + a_8 + a_{13} + a_{14} \]
Substituting the expressions:
\[ (a_1 + d) + (a_1 + 2d) + (a_1 + 7d) + (a_1 + 12d) + (a_1 + 13d) \]
Combining like terms:
\[ 5a_1 + (1 + 2 + 7 + 12 + 13)d = 5a_1 + 35d \]

6. **Using Equation 1**:
From Equation 1, we have:
\[ a_1 + 7d = 5 \implies 5a_1 + 35d = 5 \cdot 5 = 25 \]

7. **Final Result**:
Therefore,
\[ a_2 + a_3 + a_8 + a_{13} + a_{14} = 25 \]

### Conclusion:
The value of \( a_2 + a_3 + a_8 + a_{13} + a_{14} \) is \( \boxed{25} \).