Find the number of terms in each of the following A.P.s : `18,15 1/2 , 13 , `……..` -47`

Options

A. 27
B. 28
C. 29
D. 17

Answer

To find the number of terms in the given arithmetic progression (A.P.) \( 18, 15 \frac{1}{2}, 13, \ldots, -47 \), we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d).
- The first term \( a = 18 \).
- The second term is \( 15 \frac{1}{2} = 15.5 \).
- The common difference \( d \) can be calculated as:
\[ d = \text{second term} - \text{first term} = 15.5 - 18 = -2.5 \]

### Step 2: Write the formula for the nth term of an A.P.
The nth term \( a_n \) of an A.P. can be expressed as:
\[ a_n = a + (n-1) \cdot d \]
Where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 3: Set up the equation using the last term.
We know that the last term \( a_n = -47 \). Substituting the known values into the formula:
\[ -47 = 18 + (n-1)(-2.5) \]

### Step 4: Solve for n.
Rearranging the equation:
\[ -47 - 18 = (n-1)(-2.5) \]
\[ -65 = (n-1)(-2.5) \]
Dividing both sides by -2.5:
\[ n - 1 = \frac{-65}{-2.5} \]
Calculating the right-hand side:
\[ n - 1 = 26 \]
Adding 1 to both sides:
\[ n = 26 + 1 = 27 \]

### Conclusion
Thus, the total number of terms in the A.P. is \( \boxed{27} \).