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) Given,

7, 13, 19, ……., 205.

In the above A.P.,

First term (a) = 7 and common difference (d) = 13 - 7 = 6.

Let nth term be 205.

By formula,

a_n = a + (n - 1)d

Substituting values we get :

⇒ 205 = 7 + 6(n - 1)

⇒ 205 = 7 + 6n - 6

⇒ 205 = 6n + 1

⇒ 6n = 205 - 1

⇒ 6n = 204

⇒ n = $\dfrac{204}{6}$ = 34.

Hence, number of terms = 34.

(ii) Given,

18, $15\dfrac{1}{2}, 13, ……., -47$

In the above A.P.,

First term (a) = 18,

Common difference (d) = $15\dfrac{1}{2} - 18 = \dfrac{31}{2} - 18 = \dfrac{31 - 36}{2} = -\dfrac{5}{2}$.

Let nth term be -47.

By formula,

a_n = a + (n - 1)d

Substituting values we get :

⇒ -47 = $18 + (n - 1) \times -\dfrac{5}{2}$

⇒ -47 = $18 + \dfrac{-5n + 5}{2}$

⇒ -47 = $\dfrac{36 - 5n + 5}{2}$

⇒ -94 = 41 - 5n

⇒ 5n = 41 + 94

⇒ 5n = 135

⇒ n = $\dfrac{135}{5}$ = 27.

Hence, no. of terms = 27.