Question

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

Answer 1

The given AP is 18, $15\frac{1}{2}$, 13, ..., −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,

$$\begin{gathered} a_n=-47 \\ \Rightarrow 18+\left(n-1\right)\times \left(-\frac{5}{2}\right)=-47\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow -\frac{5}{2}\left(n-1\right)=-47-18=-65 \\ \Rightarrow n-1=-65\times \left(-\frac{2}{5}\right)=26 \end{gathered}$$
$\Rightarrow n=26+1=27$

Hence, there are 27 terms in the given AP.