An A.P. consists of 57 terms of which 7^th term is 13 and the last term is 138. Find the 45^th term of this A.P.

Number of terms in A.P. (n) = 57

Given,

We know that,

a_n = a + (n - 1)d

⇒ a₇ = a + 6d

⇒ a + 6d = 13 ….(1)

⇒ a₅₇ = a + 56d

⇒ a + 56d = 138 ….(2)

Subtracting equation (1) from equation (2), we get:

⇒ (a + 56d) - (a + 6d) = 138 - 13

⇒ 50d = 125

⇒ d = $\dfrac{125}{50}$

⇒ d = 2.5

Substituting value of d in equation (1), we get :

⇒ a + 6(2.5) = 13

⇒ a + 15 = 13

⇒ a = 13 - 15

⇒ a = -2

45th term:

⇒ a₄₅ = a + (45 - 1)d

⇒ a₄₅ = -2 + (45 - 1)(2.5)

⇒ a₄₅ = -2 + 110

⇒ a₄₅ = 108.

Hence, the 45^th term of this A.P. = 108.