Find the sum of the first 12 terms of the A.P. -37, -33, -29…….. .

Answer

To find the sum of the first 12 terms of the arithmetic progression (A.P.) given as -37, -33, -29, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d).
The first term \( a \) is the first term of the A.P.:
\[ a = -37 \]

To find the common difference \( d \), we subtract the first term from the second term:
\[ d = -33 - (-37) = -33 + 37 = 4 \]

### 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 \( S_{12} \) (the sum of the first 12 terms), so \( n = 12 \):
\[ S_{12} = \frac{12}{2} \times (2(-37) + (12 - 1) \cdot 4) \]

### Step 4: Simplify the expression.
Calculating inside the parentheses:
1. Calculate \( 2a \):
\[ 2a = 2 \times -37 = -74 \]

2. Calculate \( n - 1 \):
\[ n - 1 = 12 - 1 = 11 \]

3. Calculate \( (n - 1)d \):
\[ (n - 1)d = 11 \cdot 4 = 44 \]

Now substitute these values back into the equation:
\[ S_{12} = 6 \times (-74 + 44) \]

### Step 5: Final calculation.
Calculate \( -74 + 44 \):
\[ -74 + 44 = -30 \]

Now, substitute this back:
\[ S_{12} = 6 \times -30 = -180 \]

### Conclusion:
The sum of the first 12 terms of the A.P. is:
\[ S_{12} = -180 \]

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