The sum of an A.P. is 525. If its first term is 3 and the last term is 39 then its common difference is

Options

A. `3//2`
B. 1
C. `1//2`
D. none of these

Answer

To find the common difference of the arithmetic progression (A.P.) given the sum, first term, and last term, we can follow these steps:

### Step 1: Identify the given values
- First term (a) = 3
- Last term (l) = 39
- Sum of the A.P. (S_n) = 525

### 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:

\[ S_n = \frac{n}{2} \times (a + l) \]

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:

\[ 525 = \frac{n}{2} \times (3 + 39) \]

### Step 4: Simplify the equation
Calculate \(3 + 39\):

\[ 3 + 39 = 42 \]

Now substitute this back into the equation:

\[ 525 = \frac{n}{2} \times 42 \]

### Step 5: Solve for n
Multiply both sides by 2 to eliminate the fraction:

\[ 1050 = n \times 42 \]

Now, divide both sides by 42:

\[ n = \frac{1050}{42} = 25 \]

### Step 6: Use the formula for the last term of an A.P.
The formula for the last term of an A.P. is given by:

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

### Step 7: Substitute the known values into this formula
Substituting the known values:

\[ 39 = 3 + (25 - 1) \times d \]

### Step 8: Simplify the equation
Calculate \(25 - 1\):

\[ 39 = 3 + 24d \]

Now, subtract 3 from both sides:

\[ 36 = 24d \]

### Step 9: Solve for d
Divide both sides by 24:

\[ d = \frac{36}{24} = \frac{3}{2} \]

### Final Answer
The common difference \(d\) is \(\frac{3}{2}\).

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