The first term of an Arithmetic Progression is 22 and the least term is -11. If the sum is 66, the number of terms in the sequence is

Options

A. A)10
B. B)12
C. C)9
D. D)8

Answer

To solve the problem step by step, we will use the formula for the sum of an arithmetic progression (AP).

### Given:
- First term (a) = 22
- Last term (l) = -11
- Sum (S) = 66

### Step 1: Use the formula for the sum of an AP
The formula for the sum of the first n terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \times (a + l) \]
Where:
- \( S_n \) = sum of the first n terms
- \( n \) = number of terms
- \( a \) = first term
- \( l \) = last term

### Step 2: Substitute the known values into the formula
We can substitute the known values into the formula:
\[ 66 = \frac{n}{2} \times (22 + (-11)) \]
This simplifies to:
\[ 66 = \frac{n}{2} \times (22 - 11) \]
\[ 66 = \frac{n}{2} \times 11 \]

### Step 3: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 66 \times 2 = n \times 11 \]
\[ 132 = n \times 11 \]

### Step 4: Solve for n
Now, divide both sides by 11:
\[ n = \frac{132}{11} \]
\[ n = 12 \]

### Conclusion
The number of terms in the sequence is \( n = 12 \).