Find the 31st term of an A.P. whose 11th term is 38 and 6th term is 73.

Given,
a₁₁ = 38     (Eq 1)
a₆ = 73       (Eq 2)

By using formula a_n = a + (n - 1)d for Eq 1 we get,

⇒ a₁₁ = a + (11 - 1)d = 38
⇒ a + 10d = 38
⇒ a = 38 - 10d              (Eq 3)

By using formula a_n = a + (n - 1)d for Eq 2 we get,

⇒ a₆ = a + (6 - 1)d = 73
⇒ a + 5d = 73                (Eq 4)

Putting value of a from Eq 3 in Eq 4 we get,

⇒ a + 5d = 73
⇒ 38 - 10d + 5d = 73
⇒ 38 - 5d = 73
⇒ 5d = 38 - 73
⇒ 5d = -35
⇒ d = -7.

∴ a = 38 - 10d = 38 - 10(-7) = 38 + 70 = 108.

and 31st term = a₃₁ = a + (31 - 1)d = 108 + 30(-7) = 108 - 210 = -102.

Hence, the 31st term of the A.P. is -102.