What is the sum of the first 17 terms of an arithmetic progressio n if the first term is -20 and last term is 28

Options

A. 68
B. 156
C. 142
D. 242

Answer

To find the sum of the first 17 terms of an arithmetic progression (AP) where the first term \( a = -20 \) and the last term \( l = 28 \), we can follow these steps:

### Step 1: Identify the number of terms
We know that the number of terms \( n = 17 \).

### 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:
\[ 28 = -20 + (17 - 1) \cdot d \]

### Step 3: Simplify the equation
This simplifies to:
\[ 28 = -20 + 16d \]
Adding 20 to both sides:
\[ 28 + 20 = 16d \]
\[ 48 = 16d \]

### Step 4: Solve for the common difference \( d \)
Now, divide both sides by 16:
\[ d = \frac{48}{16} = 3 \]

### Step 5: 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:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
Substituting the values we have:
\[ S_{17} = \frac{17}{2} \cdot (2 \cdot (-20) + (17 - 1) \cdot 3) \]

### Step 6: Calculate the sum
Calculating inside the parentheses:
\[ S_{17} = \frac{17}{2} \cdot (2 \cdot -20 + 16 \cdot 3) \]
\[ = \frac{17}{2} \cdot (-40 + 48) \]
\[ = \frac{17}{2} \cdot 8 \]
\[ = \frac{17 \cdot 8}{2} \]
\[ = \frac{136}{2} = 68 \]

### Final Answer
Thus, the sum of the first 17 terms of the arithmetic progression is \( \boxed{68} \).

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