If the n^th term of the A.P. 58, 60, 62, …… is equal to the n^th term of the A.P. -2, 5, 12, ……., find the value of n.

n^th term of the A.P. 58, 60, 62, ……

In above sequence,

60 - 58 = 2 and 62 - 60 = 2.

Hence, above sequence is an A.P. with common difference = 2.

a_n = a + (n - 1)d

= 58 + (n - 1)2

= 58 + 2n - 2

= 2n + 56.

n^th term of the A.P. -2, 5, 12, …….

In above sequence,

5 - (-2) = 7 and 12 - 5 = 7.

Hence, above sequence is an A.P. with common difference = 7.

a_n = a + (n - 1)d

= -2 + (n - 1)7

= -2 + 7n - 7

= 7n - 9.

Since, n^th term of the A.P. 58, 60, 62, …… is equal to the n^th term of the A.P. -2, 5, 12, …….

∴ 2n + 56 = 7n - 9

⇒ 7n - 2n = 56 + 9

⇒ 5n = 65

⇒ n = 13.

Hence, n = 13.