If `a_1, a_2 ,a_3 ,......a_(24_` are in arithmetic progression and `a_1 +a_5 + a_(10) +a_(15)+a_(20)+a_(24)=225` then `a_1+a_2+a_3+.....+a_(23)+a_(24)` =

Options

A. a. 999
B. b. 900
C. c. 1225
D. d. none of these

Answer

To solve the problem step by step, we start by understanding the properties of an arithmetic progression (AP).

### Step 1: Define the terms in AP
Let \( a_1 = a \) (the first term) and the common difference be \( d \). The terms can be expressed as:
- \( a_1 = a \)
- \( a_5 = a + 4d \)
- \( a_{10} = a + 9d \)
- \( a_{15} = a + 14d \)
- \( a_{20} = a + 19d \)
- \( a_{24} = a + 23d \)

### Step 2: Write the equation based on the given information
According to the problem, we have:
\[ a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{24} = 225 \]
Substituting the expressions from Step 1:
\[ a + (a + 4d) + (a + 9d) + (a + 14d) + (a + 19d) + (a + 23d) = 225 \]

### Step 3: Combine like terms
Combining the terms, we get:
\[ 6a + (4d + 9d + 14d + 19d + 23d) = 225 \]
Calculating the sum of the coefficients of \( d \):
\[ 4 + 9 + 14 + 19 + 23 = 69 \]
Thus, the equation simplifies to:
\[ 6a + 69d = 225 \]

### Step 4: Simplify the equation
We can divide the entire equation by 3 to simplify:
\[ 2a + 23d = 75 \quad \text{(Equation 1)} \]

### Step 5: Find the sum of the first 24 terms
The sum \( S_n \) of the first \( n \) terms of an AP is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
For \( n = 24 \):
\[ S_{24} = \frac{24}{2} \times (2a + (24 - 1)d) = 12 \times (2a + 23d) \]

### Step 6: Substitute Equation 1 into the sum formula
From Equation 1, we know:
\[ 2a + 23d = 75 \]
Substituting this into the sum formula:
\[ S_{24} = 12 \times 75 = 900 \]

### Conclusion
Thus, the sum of the first 24 terms \( a_1 + a_2 + a_3 + \ldots + a_{24} \) is:
\[ \boxed{900} \]

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