Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.

Let first term be a and common difference be d.

By formula,

a_n = a + (n - 1)d

Given,

⇒ a₁₁ = 38

⇒ a + (11 - 1)d = 38

⇒ a + 10d = 38 …….(1)

Given,

⇒ a₁₆ = 73

⇒ a + (16 - 1)d = 73

⇒ a + 15d = 73 ……..(2)

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

⇒ a + 15d - (a + 10d) = 73 - 38

⇒ a - a + 15d - 10d = 35

⇒ 5d = 35

⇒ d = $\dfrac{35}{5}$

⇒ d = 7.

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

⇒ a + 10 × 7 = 38

⇒ a + 70 = 38

⇒ a = 38 - 70 = -32.

31st term = a₃₁

= a + (31 - 1)d

= -32 + 30 × 7

= -32 + 210 = 178.

Hence, 31st term of A.P. = 178.