If `a_(1)+a_(5)+a_(10)+a_(15)+a_(20)=225`, then the sum of the first 24 terms of the arithmetic progression `a_(1), a_(2), a_(3)……..` is equal to

Options

A. 450
B. 675
C. 900
D. 1200

Answer

To solve the problem, we need to find the sum of the first 24 terms of the arithmetic progression (AP) given that \( a_1 + a_5 + a_{10} + a_{15} + a_{20} = 225 \).

### Step-by-Step Solution:

1. **Identify the terms of the AP**:
- The first term of the AP is \( a_1 = a \).
- The \( n \)-th term of an AP can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
- Therefore, we can express the required terms:
- \( a_5 = a + 4d \)
- \( a_{10} = a + 9d \)
- \( a_{15} = a + 14d \)
- \( a_{20} = a + 19d \)

2. **Set up the equation**:
- Substitute the expressions for \( a_1, a_5, a_{10}, a_{15}, a_{20} \) into the given equation:
\[ a + (a + 4d) + (a + 9d) + (a + 14d) + (a + 19d) = 225 \]
- Simplifying this gives:
\[ 5a + (4d + 9d + 14d + 19d) = 225 \]
- Combine like terms:
\[ 5a + 46d = 225 \]

3. **Find the sum of the first 24 terms**:
- The formula for the sum of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) \]
- For \( n = 24 \):
\[ S_{24} = \frac{24}{2} \cdot (2a + 23d) = 12 \cdot (2a + 23d) \]

4. **Express \( S_{24} \) in terms of the previous equation**:
- From the equation \( 5a + 46d = 225 \), we can express \( 2a + 23d \):
- Multiply the equation by \( 2 \):
\[ 10a + 92d = 450 \]
- Now, we can express \( 2a + 23d \):
\[ 2a + 23d = \frac{10a + 92d}{5} = \frac{450}{5} = 90 \]

5. **Calculate \( S_{24} \)**:
- Substitute \( 2a + 23d = 90 \) into the sum formula:
\[ S_{24} = 12 \cdot 90 = 1080 \]

### Final Answer:
The sum of the first 24 terms of the arithmetic progression is \( \boxed{1080} \).