Determine the value of `S_(n)` in A.P. if we have the following:
`a=17/2, d=3/2, n=64`

Answer

To determine the value of \( S_n \) in an arithmetic progression (A.P.) given the first term \( a = \frac{17}{2} \), common difference \( d = \frac{3}{2} \), and number of terms \( n = 64 \), we can use the formula for the sum of the first \( n \) terms of an A.P.:

\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step-by-Step Solution:

1. **Identify the values:**
- First term \( a = \frac{17}{2} \)
- Common difference \( d = \frac{3}{2} \)
- Number of terms \( n = 64 \)

2. **Substitute the values into the formula:**
\[ S_n = \frac{64}{2} \times \left(2 \times \frac{17}{2} + (64 - 1) \times \frac{3}{2}\right) \]

3. **Calculate \( \frac{64}{2} \):**
\[ \frac{64}{2} = 32 \]

4. **Calculate \( 2a \):**
\[ 2a = 2 \times \frac{17}{2} = 17 \]

5. **Calculate \( n - 1 \):**
\[ n - 1 = 64 - 1 = 63 \]

6. **Calculate \( (n - 1)d \):**
\[ (n - 1)d = 63 \times \frac{3}{2} = \frac{189}{2} \]

7. **Combine the results:**
\[ S_n = 32 \times \left(17 + \frac{189}{2}\right) \]

8. **Convert 17 to a fraction with a denominator of 2:**
\[ 17 = \frac{34}{2} \]

9. **Add the fractions:**
\[ 17 + \frac{189}{2} = \frac{34}{2} + \frac{189}{2} = \frac{223}{2} \]

10. **Calculate \( S_n \):**
\[ S_n = 32 \times \frac{223}{2} = 16 \times 223 = 3568 \]

### Final Answer:
The value of \( S_n \) is \( 3568 \).

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