Find the sum of the following A.P.s :
`-37,-33,-29,……` to 12 terms .

Answer

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

### Step 1: Identify the first term and common difference
- The first term \( a \) of the A.P. is \( -37 \).
- 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 \( S_n \) of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
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, \( n = 12 \), \( a = -37 \), and \( d = 4 \). Plugging these values into the formula gives:
\[ S_{12} = \frac{12}{2} \times (2 \times -37 + (12 - 1) \times 4) \]
Calculating further:
\[ S_{12} = 6 \times (2 \times -37 + 11 \times 4) \]

### Step 4: Calculate the expression inside the parentheses
Calculating \( 2 \times -37 \):
\[ 2 \times -37 = -74 \]
Now calculating \( 11 \times 4 \):
\[ 11 \times 4 = 44 \]
Now substituting these values back:
\[ S_{12} = 6 \times (-74 + 44) \]
Calculating the sum inside the parentheses:
\[ -74 + 44 = -30 \]

### Step 5: Final calculation of the sum
Now we multiply:
\[ S_{12} = 6 \times -30 = -180 \]

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