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

Solution:

Given,

11^th term of an AP = a₁₁ = 38

and 16^th term, a₁₆ = 73

We know that,

a_n = a + (n − 1)d

a₁₁ = a + (11 − 1)d

38 = a + 10d ….(1)

In the same way,

a ₁₆ = a + (16 − 1) d

73 = a + 15d ….(2)

On subtracting equation (1) from (2), we get;

35 = 5 d

d = 7

From equation (1),

38 = a + 10 × (7)

38 − 70 = a

a = −32

a₃₁ = a + (31 − 1) d

= −32 + 30 (7)

= −32 + 210

= 178

Hence, the 31^st term of the AP is 178.