The first term of an A.P. is 3.The sum of first 25 terms is equal to the sum of next 15 terms.Find the common difference of the A.P.

Options

A. `1/4`
B. `1/6`
C. `1/9`
D. `1/2`

Answer

To solve the problem step by step, we will use the formula for the sum of an arithmetic progression (A.P.).

### Step 1: Write down the given information
- First term \( a = 3 \)
- The sum of the first 25 terms is equal to the sum of the next 15 terms.

### Step 2: Use the formula for the sum of the first \( n \) terms of an A.P.
The formula for the sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

### Step 3: Calculate the sum of the first 25 terms
Using the formula for \( n = 25 \):
\[ S_{25} = \frac{25}{2} \times (2 \times 3 + (25 - 1)d) = \frac{25}{2} \times (6 + 24d) \]
\[ S_{25} = \frac{25}{2} \times (6 + 24d) = \frac{25 \times (6 + 24d)}{2} \]

### Step 4: Calculate the sum of the next 15 terms
The next 15 terms start from the 26th term to the 40th term. The 26th term can be calculated as:
\[ a + 25d = 3 + 25d \]
The 40th term can be calculated as:
\[ a + 39d = 3 + 39d \]
Thus, the sum of the next 15 terms is:
\[ S_{15} = \frac{15}{2} \times ((3 + 25d) + (3 + 39d)) = \frac{15}{2} \times (6 + 64d) \]
\[ S_{15} = \frac{15 \times (6 + 64d)}{2} \]

### Step 5: Set the two sums equal to each other
Since the sum of the first 25 terms is equal to the sum of the next 15 terms:
\[ \frac{25 \times (6 + 24d)}{2} = \frac{15 \times (6 + 64d)}{2} \]
We can cancel the \( \frac{1}{2} \) from both sides:
\[ 25 \times (6 + 24d) = 15 \times (6 + 64d) \]

### Step 6: Expand both sides
Expanding both sides gives:
\[ 150 + 600d = 90 + 960d \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate \( d \):
\[ 150 - 90 = 960d - 600d \]
\[ 60 = 360d \]

### Step 8: Solve for \( d \)
Dividing both sides by 360:
\[ d = \frac{60}{360} = \frac{1}{6} \]

### Final Answer
The common difference \( d \) of the A.P. is \( \frac{1}{6} \).

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