Find the number of terms in each of the following APs: (i) 7, 13, 19, ..., 205 (ii) 18, 15 1/2, 13, ..., −47

(i) a = 7
d = a₂ - a₁ = 13 - 7 = 6
Considering there are n terms in this A.P.
an = 205
We know that an = 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 = a₂ - a₁ = 15 1/2 - 18 => d = 31/2 - 36/2 = -5/2
Considering there are n terms in this A.P.
an = -47
We know, an = a + (n - 1)d
-47 = 18 + (n - 1)(-5/2)
=> -47 - 18 = (n - 1)(-5/2)
=> -65 = (n - 1)(-5/2)
=> n - 1 = -65 × (-2/5) = 26
=> n = 27
The sequence has 27 terms.