Determine the number of terms in the A.P. `17, 14 1/2, 12, …………, -38`.

Answer

To determine the number of terms in the arithmetic progression (A.P.) given by the sequence \( 17, 14 \frac{1}{2}, 12, \ldots, -38 \), we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a_1 \) of the A.P. is:
\[ a_1 = 17 \]

To find the common difference \( d \), we can calculate it using the first two terms:
\[ d = a_2 - a_1 = 14 \frac{1}{2} - 17 \]
Converting \( 14 \frac{1}{2} \) to an improper fraction:
\[ 14 \frac{1}{2} = \frac{29}{2} \]
Now, substituting this back into the equation for \( d \):
\[ d = \frac{29}{2} - 17 = \frac{29}{2} - \frac{34}{2} = \frac{29 - 34}{2} = \frac{-5}{2} \]

### Step 2: Identify the last term
The last term \( l \) of the A.P. is given as:
\[ l = -38 \]

### Step 3: Use the formula for the nth term of an A.P.
The formula for the nth term \( a_n \) of an A.P. is given by:
\[ a_n = a_1 + (n - 1) d \]
Substituting the known values:
\[ -38 = 17 + (n - 1) \left(-\frac{5}{2}\right) \]

### Step 4: Solve for \( n \)
Rearranging the equation:
\[ -38 - 17 = (n - 1) \left(-\frac{5}{2}\right) \]
This simplifies to:
\[ -55 = (n - 1) \left(-\frac{5}{2}\right) \]
Multiplying both sides by \(-2\) to eliminate the fraction:
\[ 110 = 5(n - 1) \]
Dividing both sides by 5:
\[ 22 = n - 1 \]
Adding 1 to both sides:
\[ n = 23 \]

### Conclusion
Thus, the number of terms in the arithmetic progression is:
\[ \boxed{23} \]