Find `S_(3)` for the A.P. 3,5,7,9…..

Answer

To find the sum \( S_3 \) for the arithmetic progression (A.P.) given as 3, 5, 7, 9, ..., we will follow these steps:

### Step 1: Identify the first three terms of the A.P.
The first three terms of the A.P. are:
- \( a_1 = 3 \)
- \( a_2 = 5 \)
- \( a_3 = 7 \)

### Step 2: Write down the formula for the sum of the first \( n \) terms of an A.P.
The formula for the sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 3: Determine the values of \( a \), \( d \), and \( n \).
From the A.P.:
- The first term \( a = 3 \)
- The common difference \( d = 5 - 3 = 2 \)
- The number of terms \( n = 3 \)

### Step 4: Substitute the values into the formula.
Now, substituting the values into the formula for \( S_3 \):
\[ S_3 = \frac{3}{2} \times (2 \cdot 3 + (3 - 1) \cdot 2) \]
\[ = \frac{3}{2} \times (6 + 4) \]
\[ = \frac{3}{2} \times 10 \]
\[ = \frac{30}{2} = 15 \]

### Step 5: Conclusion
Thus, the sum \( S_3 \) of the first three terms of the A.P. is:
\[ S_3 = 15 \]

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