How many terms are there in the AP 18,

Question:

How many terms are there in the AP $18,15 \frac{1}{2}, 13, \ldots,-47 ?$

Solution:

The given $\mathrm{AP}$ is $18,15 \frac{1}{2}, 13, \ldots,-47$.

First term, a = 18

Common difference, $d=15 \frac{1}{2}-18=\frac{31}{2}-18=\frac{31-36}{2}=-\frac{5}{2}$

Suppose there are n terms in the given AP. Then,

$a_{n}=-47$

$\Rightarrow 18+(n-1) \times\left(-\frac{5}{2}\right)=-47 \quad\left[a_{n}=a+(n-1) d\right]$

$\Rightarrow-\frac{5}{2}(n-1)=-47-18=-65$

$\Rightarrow n-1=-65 \times\left(-\frac{2}{5}\right)=26$

$\Rightarrow n=26+1=27$

Hence, there are 27 terms in the given AP.