To find the number of terms in the arithmetic progression (A.P.) given by 17 , 14 1 / 2 , 12 , … , − 38

 

To find the number of terms in the arithmetic progression (A.P.) given by $17, 14 \frac{1}{2}, 12, \ldots, -38$, we start by identifying the first term and the common difference.

The first term ($a$) is:

$$a = 17$$

The second term is $14 \frac{1}{2} = 14.5$. Therefore, the common difference ($d$) is:

$$d = 14.5 - 17 = -2.5$$

We need to find the number of terms ($n$) for which the last term ($a_n$) is $-38$. The general formula for the $n$-th term of an A.P. is:

$$a_n = a + (n-1)d$$

Given $a_n = -38$, $a = 17$, and $d = -2.5$, we substitute these values into the formula:

$$-38 = 17 + (n-1)(-2.5)$$

Solving for $n$, we first isolate the term involving $n$:

$$-38 = 17 - 2.5(n-1)$$ $$-38 = 17 - 2.5n + 2.5$$ $$-38 = 19.5 - 2.5n$$

Next, we solve for $n$:

$$-38 - 19.5 = -2.5n$$ $$-57.5 = -2.5n$$ $$n = \frac{57.5}{2.5}$$ $$n = 23$$

Therefore, the number of terms in the arithmetic progression is:

$$\boxed{23}$$