What to the sum of the first 13 terms of an arithmetic progression if the first term to -10 and last term is 26?

Options

A. 104
B. 140
C. 84
D. 98

Answer

To find the sum of the first 13 terms of an arithmetic progression (AP) where the first term (a) is -10 and the last term (l) is 26, we can use the formula for the sum of the first n terms of an AP.

### Step-by-Step Solution:

1. **Identify the given values:**
- First term (a) = -10
- Last term (l) = 26
- Number of terms (n) = 13

2. **Use the formula for the sum of the first n terms of an AP:**
The sum \( S_n \) can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a + l) \]
where:
- \( S_n \) = sum of the first n terms
- \( n \) = number of terms
- \( a \) = first term
- \( l \) = last term

3. **Substitute the values into the formula:**
\[ S_{13} = \frac{13}{2} \times (-10 + 26) \]

4. **Calculate the expression inside the parentheses:**
\[ -10 + 26 = 16 \]

5. **Now substitute this back into the sum formula:**
\[ S_{13} = \frac{13}{2} \times 16 \]

6. **Calculate \( \frac{13}{2} \times 16 \):**
\[ S_{13} = \frac{13 \times 16}{2} = \frac{208}{2} = 104 \]

7. **Final result:**
The sum of the first 13 terms of the arithmetic progression is \( S_{13} = 104 \).