Find the sum of 11 terms of -7, -2, 3 , 8 ,…..

Answer

To find the sum of the first 11 terms of the sequence -7, -2, 3, 8, ..., we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \( a \) of the sequence is:
\[ a = -7 \]

To find the common difference \( d \), we can subtract the first term from the second term:
\[ d = -2 - (-7) = -2 + 7 = 5 \]

### Step 2: Use the formula for the sum of the first \( n \) terms of an arithmetic series
The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series is:
\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 3: Substitute the values into the formula
Here, we need to find the sum of the first 11 terms, so \( n = 11 \):
\[ S_{11} = \frac{11}{2} \left(2(-7) + (11 - 1)(5)\right) \]

### Step 4: Simplify the expression inside the parentheses
Calculating \( 2a \):
\[ 2a = 2 \times -7 = -14 \]

Calculating \( (n - 1)d \):
\[ (n - 1)d = 10 \times 5 = 50 \]

Now, substitute these values back into the equation:
\[ S_{11} = \frac{11}{2} \left(-14 + 50\right) \]

### Step 5: Calculate the sum
Now, simplify the expression:
\[ -14 + 50 = 36 \]
So,
\[ S_{11} = \frac{11}{2} \times 36 \]

Calculating \( \frac{11 \times 36}{2} \):
\[ \frac{11 \times 36}{2} = \frac{396}{2} = 198 \]

### Final Answer
The sum of the first 11 terms is:
\[ S_{11} = 198 \]

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