In an A.P., a = 3, l = 39 and S = 525, then the value of common difference d =

Options

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

Answer

To find the common difference \( d \) in the arithmetic progression (A.P.) where the first term \( a = 3 \), the last term \( l = 39 \), and the sum \( S = 525 \), we can follow these steps:

### Step 1: Use the formula for the sum of the first \( n \) terms of an A.P.

The formula for the sum \( S \) of the first \( n \) terms of an A.P. is given by:

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

### Step 2: Substitute the known values into the formula.

We know:
- \( S = 525 \)
- \( a = 3 \)
- \( l = 39 \)

Substituting these values into the sum formula:

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

### Step 3: Simplify the equation.

Calculate \( 3 + 39 \):

\[ 3 + 39 = 42 \]

Now substitute back into the equation:

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

### Step 4: 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 5: Use the formula for the last term of an A.P.

The last term \( l \) can also be expressed as:

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

Substituting the known values:

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

### Step 6: Simplify and solve for \( d \).

This simplifies to:

\[ 39 = 3 + 24d \]

Subtract 3 from both sides:

\[ 36 = 24d \]

Now, divide both sides by 24:

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

### Final Answer:

The value of the common difference \( d \) is \( \frac{3}{2} \).

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