Find the value of d, when `S_(n) = 240, n =10 , and a=6`.

Answer

To find the value of \( d \) given \( S_n = 240 \), \( n = 10 \), and \( a = 6 \), we will use the formula for the sum of the first \( n \) terms of an arithmetic series:

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

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

We know:
- \( S_n = 240 \)
- \( n = 10 \)
- \( a = 6 \)

Substituting these values into the formula gives us:

\[ 240 = \frac{10}{2} \times (2 \times 6 + (10 - 1)d) \]

### Step 2: Simplify the equation

Calculating \( \frac{10}{2} \):

\[ 240 = 5 \times (12 + 9d) \]

### Step 3: Divide both sides by 5

To isolate the term with \( d \), divide both sides by 5:

\[ \frac{240}{5} = 12 + 9d \]

This simplifies to:

\[ 48 = 12 + 9d \]

### Step 4: Subtract 12 from both sides

Now, we will subtract 12 from both sides to solve for \( d \):

\[ 48 - 12 = 9d \]

This gives us:

\[ 36 = 9d \]

### Step 5: Divide both sides by 9

Finally, we divide both sides by 9 to find \( d \):

\[ d = \frac{36}{9} \]

This simplifies to:

\[ d = 4 \]

### Final Answer

The value of \( d \) is \( 4 \).

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