What is the sum of the first 13 terms of an arithmetic progression if the first term is -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 follow these steps:

### Step 1: Identify the number of terms (n)
We are given that the first term is -10 and we need to find the sum of the first 13 terms. Therefore, \( n = 13 \).

### Step 2: Use the formula for the last term of an AP
The formula for the last term of an arithmetic progression is given by:
\[ l = a + (n - 1) \cdot d \]
Where:
- \( l \) is the last term,
- \( a \) is the first term,
- \( n \) is the number of terms,
- \( d \) is the common difference.

Substituting the known values:
\[ 26 = -10 + (13 - 1) \cdot d \]

### Step 3: Solve for the common difference (d)
Rearranging the equation:
\[ 26 = -10 + 12d \]
Adding 10 to both sides:
\[ 36 = 12d \]
Dividing both sides by 12:
\[ d = 3 \]

### Step 4: Use the formula for the sum of the first n terms of an AP
The formula for the sum \( S_n \) of the first n terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]

### Step 5: Substitute the values into the sum formula
Substituting \( n = 13 \), \( a = -10 \), and \( d = 3 \):
\[ S_{13} = \frac{13}{2} \cdot (2 \cdot -10 + (13 - 1) \cdot 3) \]
Calculating the expression inside the parentheses:
\[ S_{13} = \frac{13}{2} \cdot (-20 + 12 \cdot 3) \]
Calculating \( 12 \cdot 3 = 36 \):
\[ S_{13} = \frac{13}{2} \cdot (-20 + 36) \]
Calculating \( -20 + 36 = 16 \):
\[ S_{13} = \frac{13}{2} \cdot 16 \]
Calculating \( \frac{13 \cdot 16}{2} = \frac{208}{2} = 104 \)

### Final Answer
The sum of the first 13 terms of the arithmetic progression is \( \boxed{104} \).