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.).

### Step 1: Write down the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first n terms (S_n) of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 2: Substitute the known values into the formula
From the question, we know:
- \( a = 3 \) (the first term)
- \( n = 8 \) (the number of terms)
- \( S_n = 192 \) (the sum of the first n terms)

Substituting these values into the formula gives:
\[ 192 = \frac{8}{2} \times (2 \cdot 3 + (8 - 1)d) \]

### Step 3: Simplify the equation
First, simplify \( \frac{8}{2} \):
\[ 192 = 4 \times (6 + 7d) \]

Now, divide both sides by 4:
\[ 48 = 6 + 7d \]

### Step 4: Isolate the term with d
Subtract 6 from both sides:
\[ 48 - 6 = 7d \]
\[ 42 = 7d \]

### Step 5: Solve for d
Now, divide both sides by 7:
\[ d = \frac{42}{7} \]
\[ d = 6 \]

### Final Answer
The common difference \( d \) is 6.

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