If t_n represents n^th term of an A.P., t₂ + t₅ - t₃ = 10 and t₂ + t₉ = 17, find its first term and its common difference.

t_n = a + (n - 1)d …….(1)

Substituting above value in t₂ + t₅ - t₃ = 10,

⇒ [a + (2 - 1)d] + [a + (5 - 1)d] - [a + (3 - 1)d] = 10

⇒ (a + d) + (a + 4d) - (a + 2d) = 10

⇒ a + 3d = 10 …….(i)

Substituting value from 1 in t₂ + t₉ = 17,

⇒ [a + (2 - 1)d] + [a + (9 - 1)d] = 17

⇒ a + d + a + 8d = 17

⇒ 2a + 9d = 17 ……..(ii)

Multiplying (i) by 2 and then subtracting from (ii) we get,

⇒ 2a + 9d - 2(a + 3d) = 17 - 2(10)

⇒ 2a + 9d - 2a - 6d = 17 - 20

⇒ 3d = -3

⇒ d = -1.

Substituting value of d in (i) we get,

⇒ a + 3(-1) = 10

⇒ a - 3 = 10

⇒ a = 10 + 3 = 13.

Hence, first term = 13 and common difference = -1.