If `t_r` denotes the `r^(th)` term of an A.P. and `t_1+ t_5 + t_10 + T_15 + t_20 + t_24` = 75 , then sum of its first 24 terms is :

Options

A. 400
B. 600
C. 300
D. 900

Answer

To solve the problem step by step, we will denote the first term of the arithmetic progression (A.P.) as \( a \) and the common difference as \( d \).

### Step 1: Write the terms of the A.P.
The terms can be expressed as follows:
- \( t_1 = a \)
- \( t_5 = a + 4d \)
- \( t_{10} = a + 9d \)
- \( t_{15} = a + 14d \)
- \( t_{20} = a + 19d \)
- \( t_{24} = a + 23d \)

### Step 2: Set up the equation
According to the problem, we have:
\[ t_1 + t_5 + t_{10} + t_{15} + t_{20} + t_{24} = 75 \]
Substituting the expressions for the terms:
\[ a + (a + 4d) + (a + 9d) + (a + 14d) + (a + 19d) + (a + 23d) = 75 \]

### Step 3: Simplify the equation
Combine all the terms:
\[ 6a + (4d + 9d + 14d + 19d + 23d) = 75 \]
Calculating the sum of the coefficients of \( d \):
\[ 4 + 9 + 14 + 19 + 23 = 69 \]
Thus, we have:
\[ 6a + 69d = 75 \]

### Step 4: Solve for \( a \) and \( d \)
Dividing the entire equation by 3:
\[ 2a + 23d = 25 \]

### Step 5: Express \( a + 23d \)
From the equation \( 2a + 23d = 25 \), we can express it as:
\[ a + 23d = 25 - a \]

### Step 6: Find the 24th term
The 24th term \( t_{24} \) can be expressed as:
\[ t_{24} = a + 23d \]
From the previous step, we have:
\[ t_{24} = 25 - a \]

### Step 7: Find the sum of the first 24 terms
The sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} (a + l) \]
where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

For the first 24 terms:
\[ S_{24} = \frac{24}{2} (a + t_{24}) = 12 (a + t_{24}) \]
Substituting \( t_{24} = 25 - a \):
\[ S_{24} = 12 (a + (25 - a)) = 12 \times 25 = 300 \]

### Final Answer
Thus, the sum of the first 24 terms is:
\[ \boxed{300} \]