In an A.P., a=-6 and d = 2. Find the sum of its first 20 terms.

Answer

To find the sum of the first 20 terms of an arithmetic progression (A.P.) where the first term \( a = -6 \) and the common difference \( d = 2 \), we can use the formula for the sum of the first \( n \) terms of an A.P.:

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

### Step-by-step solution:

1. **Identify the values:**
- First term \( a = -6 \)
- Common difference \( d = 2 \)
- Number of terms \( n = 20 \)

2. **Substitute the values into the formula:**
\[ S_{20} = \frac{20}{2} \cdot (2 \cdot (-6) + (20 - 1) \cdot 2) \]

3. **Calculate \( \frac{20}{2} \):**
\[ \frac{20}{2} = 10 \]

4. **Calculate \( 2a \):**
\[ 2 \cdot (-6) = -12 \]

5. **Calculate \( (n - 1) \cdot d \):**
\[ (20 - 1) \cdot 2 = 19 \cdot 2 = 38 \]

6. **Combine the results:**
\[ S_{20} = 10 \cdot (-12 + 38) \]

7. **Calculate \( -12 + 38 \):**
\[ -12 + 38 = 26 \]

8. **Final calculation:**
\[ S_{20} = 10 \cdot 26 = 260 \]

### Final answer:
The sum of the first 20 terms of the A.P. is \( S_{20} = 260 \).

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