Question

Find the number of terms in each of the following APs respectively:

$\left(i\right)7,13,\mathrm{19....},205$
$\left(ii\right)18,\frac{31}{2},13,...-47$

Answer 1

(i)
$7,13,\mathrm{19....},205$

First term, $a=7$

Common difference, $d=13-7=19-13=6$

$a_n=205$

Using formula $a_n=a+(n-1)d$ to find $n^{th}$ term of arithmetic progression, we get

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

$\Rightarrow 205=7+(n-1)6$

$\Rightarrow 205=6n+1$

$\Rightarrow 204=6n$

$\Rightarrow n=\frac{204}{6}=34$

Therefore, there are 34 terms in the given arithmetic progression.

(ii)
$18,\frac{31}{2},13,...-47$

First term, $a=18$

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

$a_n=-47$

Using formula $a_n=a+(n-1)d$ to find $n^{th}$ term of arithmetic progression, we get

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

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

$\Rightarrow -47=18-\frac{5n}{2}+\frac{5}{2}$

Multiplying by 2 on both sides, we get,

$\Rightarrow -94=36-5n+5$

$\Rightarrow 5n=94+36+5=135$

$\Rightarrow n=\frac{135}{5}=27$

Therefore, there are 27 terms in the given arithmetic progression.