Find the number of terms in each of the following APs:
(i) 7, 13, 19…., 205
(ii) 18,\({15}{\dfrac{1}{2}}\) , 13…, -47

(i) 7, 13, 19 …, 205

First term = a = 7,
Common difference = d
= 13 – 7 = 19 – 13 = 6

And \(a_n = 205\)

Using formula \(a_n = a + (n - 1)d\) , to find nth term of arithmetic progression,

\(\Rightarrow \) 205 = 7 + (n - 1) 6 = 7 + 6n – 6
\(\Rightarrow \) 205 = 6n + 1
\(\Rightarrow \) 204 = 6n
\(\Rightarrow \) n = 34

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


(ii) 18, \({15}{\dfrac{1}{2}}\) , 13 …, -47

First term = a =18,
Common difference = d
= \({15}{\dfrac{1}{2}} - 18 \)
\( = {{31} \over {2}} - 18 \)
\( = {{31 - 36} \over {2}} \)
\( = {{-5} \over {2}}\)

And \(a_n = -47\)

Using formula \(a_n = a + (n - 1)d\), to find nth term of arithmetic progression,

\(\Rightarrow \) 47 = 18 + (n - 1) \({{-5} \over {2}}\)
\(\Rightarrow \) 36 - \({{5} \over {2}}\) n + \({{5} \over {2}}\)
\(\Rightarrow\) -94 = 36 - 5n + 5
\(\Rightarrow \) 5n = 135
\(\Rightarrow \) n = 27

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