Question

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

(i) $7,13,19,....,205$

(ii) $18,15\frac{1}{2},13,...,-47$

Answer 1

(i) $a=7$

$d=a_2-a_1=13-7=6$

Considering there are $n$ terms in this A.P.

$a_n=205$

We know that $a_n=a+(n-1)d$

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

$198=(n-1)6$

$33=(n-1)$

$n=34$

The sequence has $34$ terms.

(ii) $a=18$

$d=a_2-a_1=15\frac{1}{2}-18$

$\Rightarrow d=\frac{31-36}{2}=\frac{-5}{2}$

Considering there are $n$ terms in this A.P.

$a_n=-47$

We know, $a_n=a+(n-1)d$

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

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

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

$\Rightarrow n-1=\frac{-130}{-5}=26$

$\Rightarrow n=27$

The sequence has $27$ terms.