In an A. P. :
given `a = 3, n = 8, S = 192,` find d.

Answer

To solve the problem step by step, we will use the formula for the sum of the first n terms of an arithmetic progression (A.P.). The formula is:

\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

Where:
- \( S_n \) is the sum of the first n terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

Given:
- \( a = 3 \)
- \( n = 8 \)
- \( S_n = 192 \)

### Step 1: Substitute the known values into the formula

We substitute \( S_n = 192 \), \( n = 8 \), and \( a = 3 \) into the formula:

\[ 192 = \frac{8}{2} \times (2 \times 3 + (8 - 1)d) \]

### Step 2: Simplify the equation

Calculating \( \frac{8}{2} \):

\[ 192 = 4 \times (6 + 7d) \]

### Step 3: Expand the equation

Now, we expand the equation:

\[ 192 = 24 + 28d \]

### Step 4: Isolate the term with d

Next, we isolate \( 28d \) by subtracting 24 from both sides:

\[ 192 - 24 = 28d \]

This simplifies to:

\[ 168 = 28d \]

### Step 5: Solve for d

Now, we divide both sides by 28 to find \( d \):

\[ d = \frac{168}{28} \]

Calculating this gives:

\[ d = 6 \]

### Final Answer

The common difference \( d \) is \( 6 \).

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