Find the 31^st term of an A.P. whose 10^th term is 38 and 16^th term is 74.

We know that,

n^th term of an A.P. is given by,

a_n = a + (n - 1)d

Given, 10^th term is 38

∴ a₁₀ = a + (10 - 1)d

⇒ 38 = a + 9d

⇒ a + 9d = 38 ……..(i)

Given, 16^th term is 74

∴ a₁₆ = a + (16 - 1)d

⇒ 74 = a + 15d

⇒ a + 15d = 74 ……..(ii)

Subtracting (i) from (ii) we get,

⇒ a + 15d - (a + 9d) = 74 - 38

⇒ 6d = 36

⇒ d = 6.

Substituting value of d in (i) we get,

⇒ a + 9(6) = 38

⇒ a + 54 = 38

⇒ a = -16.

31^st term = a₃₁

a₃₁ = a + (31 - 1)d

= -16 + 30(6)

= -16 + 180

= 164.

Hence, 31^st term of A.P. = 164.