If the sum of five terms of an A.P. is 75, then find the third term of the A.P.

Answer

To find the third term of an arithmetic progression (A.P.) given that the sum of five terms is 75, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the sum of the first n terms of an A.P.**:
The sum \( S_n \) of the first \( n \) terms of an A.P. can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

2. **Substitute the known values**:
In this case, we know that \( S_5 = 75 \) (the sum of five terms) and \( n = 5 \). Therefore, we can substitute these values into the formula:
\[ 75 = \frac{5}{2} \times (2a + (5-1)d) \]

3. **Simplify the equation**:
Simplifying the equation gives:
\[ 75 = \frac{5}{2} \times (2a + 4d) \]
To eliminate the fraction, multiply both sides by 2:
\[ 150 = 5 \times (2a + 4d) \]
Now divide both sides by 5:
\[ 30 = 2a + 4d \]

4. **Rearranging the equation**:
We can simplify this further by dividing the entire equation by 2:
\[ 15 = a + 2d \]

5. **Identify the third term**:
The third term \( T_3 \) of an A.P. can be expressed as:
\[ T_3 = a + 2d \]
From our previous step, we have already found that \( a + 2d = 15 \). Therefore:
\[ T_3 = 15 \]

### Conclusion:
The third term of the A.P. is \( \boxed{15} \).