Sum of first 20 terms of the Arithmetic Progression (A.P.) 11, 7, 3, _____ is

Options

A. 20
B. -540
C. 980
D. -65

Answer

To find the sum of the first 20 terms of the given Arithmetic Progression (A.P.) 11, 7, 3, ..., we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
- The first term \( a = 11 \).
- The second term is \( 7 \), so the common difference \( d = 7 - 11 = -4 \).

### Step 2: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first n terms (S_n) of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Here, \( n = 20 \).

### Step 3: Substitute the values into the formula
Substituting \( n = 20 \), \( a = 11 \), and \( d = -4 \) into the formula:
\[ S_{20} = \frac{20}{2} \times (2 \times 11 + (20 - 1) \times -4) \]

### Step 4: Simplify the expression
Calculating inside the parentheses:
\[ S_{20} = 10 \times (22 + 19 \times -4) \]
Calculating \( 19 \times -4 = -76 \):
\[ S_{20} = 10 \times (22 - 76) \]
Calculating \( 22 - 76 = -54 \):
\[ S_{20} = 10 \times -54 \]
Calculating \( 10 \times -54 = -540 \).

### Final Answer
The sum of the first 20 terms of the A.P. is:
\[ \boxed{-540} \]