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 Arithmetic Progression (A.P.) given as 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:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 3: Substitute the values into the formula
Here, we need to find the sum of the first 20 terms, so \( n = 20 \):
\[ S_{20} = \frac{20}{2} \times (2 \times 11 + (20 - 1)(-4)) \]

### Step 4: Simplify the expression
Calculating step by step:
1. Calculate \( \frac{20}{2} = 10 \).
2. Calculate \( 2 \times 11 = 22 \).
3. Calculate \( 20 - 1 = 19 \).
4. Calculate \( 19 \times -4 = -76 \).
5. Now substitute back into the equation:
\[ S_{20} = 10 \times (22 - 76) \]
6. Calculate \( 22 - 76 = -54 \).
7. Finally, calculate \( 10 \times -54 = -540 \).

### Conclusion
The sum of the first 20 terms of the A.P. is \( -540 \).