If in an A.P. `{a_(n)}`,
`a_(1)+a_(5)+a_(10)+a_(15)+_(20)+a_(24)=`225 then : `S_(24)= cdots`

Options

A. 550
B. 900
C. 1150
D. 1400

Answer

To solve the problem, we need to find the sum \( S_{24} \) of the first 24 terms of an arithmetic progression (A.P.) given that:

\[ a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{24} = 225 \]

### Step 1: Express the terms in the A.P.
In an A.P., the \( n \)-th term can be expressed as:

\[ a_n = a + (n-1)d \]

where \( a \) is the first term and \( d \) is the common difference. Therefore, we can express the terms as follows:

- \( a_1 = a \)
- \( a_5 = a + 4d \)
- \( a_{10} = a + 9d \)
- \( a_{15} = a + 14d \)
- \( a_{20} = a + 19d \)
- \( a_{24} = a + 23d \)

### Step 2: Write the equation based on the given sum
Now, substituting these expressions into the equation:

\[ a + (a + 4d) + (a + 9d) + (a + 14d) + (a + 19d) + (a + 23d) = 225 \]

### Step 3: Combine like terms
Combining the terms gives:

\[ 6a + (4d + 9d + 14d + 19d + 23d) = 225 \]

Calculating the sum of the coefficients of \( d \):

\[ 4 + 9 + 14 + 19 + 23 = 69 \]

So we have:

\[ 6a + 69d = 225 \]

### Step 4: Simplify the equation
To simplify, we can divide the entire equation by 3:

\[ 2a + 23d = 75 \]

### Step 5: Find the sum \( S_{24} \)
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by:

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

For \( n = 24 \):

\[ S_{24} = \frac{24}{2} \times (2a + 23d) = 12 \times (2a + 23d) \]

### Step 6: Substitute the value from earlier
From our earlier equation, we know:

\[ 2a + 23d = 75 \]

Substituting this into the equation for \( S_{24} \):

\[ S_{24} = 12 \times 75 = 900 \]

### Final Answer
Thus, the sum of the first 24 terms \( S_{24} \) is:

\[ \boxed{900} \]