Given : The first A.P. is 63, 65, 67, ……
where a = 63, d = a2 – a1 ,
⇒ d = 65 – 63 = 2
and the second A.P. is 3, 10, 17, …….
where a = 3; d = a2 – a1 = 10 – 3 = 7
Suppose the nth terms of the two A.Ps are equal,
where an = a + (n – 1)d
⇒ 63 + (n – 1)2 = 3 + (n – 1)7
⇒ 63 + 2n – 2 = 3 + 7n – 7
⇒ 61 + 2n = 7n – 4
⇒ 7n – 2n = 61 + 4
⇒ 5n = 65
⇒ n = 65/5 = 13
∴ 13th terms of the two A.Ps are equal.