Find the sum of the first 22 terms of the A.P. : 8, 3, -2,………….

Answer

To find the sum of the first 22 terms of the arithmetic progression (A.P.) given by the sequence 8, 3, -2, ..., we will follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) is the first number in the sequence:
\[ a = 8 \]

To find the common difference \( d \), we subtract the first term from the second term:
\[ d = 3 - 8 = -5 \]

### Step 2: Use 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 3: Substitute the values into the formula
Here, we need to find the sum of the first 22 terms, so \( n = 22 \):
\[ S_{22} = \frac{22}{2} \times (2 \times 8 + (22 - 1)(-5)) \]

### Step 4: Simplify the expression
Calculate \( \frac{22}{2} \):
\[ \frac{22}{2} = 11 \]

Now calculate \( 2 \times 8 \):
\[ 2 \times 8 = 16 \]

Next, calculate \( (22 - 1)(-5) \):
\[ (22 - 1) = 21 \]
\[ 21 \times (-5) = -105 \]

Now substitute these values back into the equation:
\[ S_{22} = 11 \times (16 - 105) \]

### Step 5: Calculate the final sum
Now calculate \( 16 - 105 \):
\[ 16 - 105 = -89 \]

Now multiply by 11:
\[ S_{22} = 11 \times (-89) = -979 \]

### Final Answer
The sum of the first 22 terms of the A.P. is:
\[ \boxed{-979} \]