If for an A.P., `t_(8) = 36`, find `S_(15)`.

Answer

To solve the problem, we need to find the sum of the first 15 terms \( S_{15} \) of an arithmetic progression (A.P.) given that the 8th term \( t_8 = 36 \).

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an A.P.**:
The nth term of an A.P. can be expressed as:
\[ t_n = a + (n - 1)d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

2. **Substitute the values for the 8th term**:
We know that \( t_8 = 36 \). Therefore, we can write:
\[ t_8 = a + (8 - 1)d = a + 7d \]
Setting this equal to 36 gives us our first equation:
\[ a + 7d = 36 \quad \text{(Equation 1)} \]

3. **Understand the formula for the sum of the first n terms of an A.P.**:
The sum of the first n terms \( S_n \) can be expressed as:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

4. **Substitute the values for the sum of the first 15 terms**:
We need to find \( S_{15} \):
\[ S_{15} = \frac{15}{2} \times (2a + (15 - 1)d) = \frac{15}{2} \times (2a + 14d) \]

5. **Factor out the common terms**:
We can factor out 2 from the expression:
\[ S_{15} = \frac{15}{2} \times 2 \times \left( a + 7d \right) = 15 \times (a + 7d) \]

6. **Substitute \( a + 7d \) from Equation 1**:
From Equation 1, we know that \( a + 7d = 36 \). Therefore:
\[ S_{15} = 15 \times 36 \]

7. **Calculate \( S_{15} \)**:
\[ S_{15} = 540 \]

### Final Answer:
\[ S_{15} = 540 \]