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

To Find : 31st term.
Given: 11th term of AP, a₁₁ = 38 and 16th term of AP,  a₁₆ = 73

We know that a_n = a + (n – 1)d

where, a_n = nth terms of AP
a = first term of AP
n = number of terms
d = common difference

Hence,
a₁₁ = a + (11 - 1)d
a₁₁ = a + 10d = 38          .......eq(i)

And,
a₁₆ = a + (16 - 1)d
a₁₆ = a + 15d = 73          ........eq(ii)

Subtracting eq(i) from eq(ii), we get following:

a + 15d – (a + 10d) = 73 – 38

a + 15d - a - 10 d = 35

Or, 5d = 35

Or, d = 7

Substituting the value of d in eq(i) we get;

a + 10 x 7 = 38

Or, a + 70 = 38

Or, a = 38 – 70 = - 32

Now 31st term can be calculated as follows:

a₃₁ = a + (31 - 1)d

a₃₁ = a + 30d

= - 32 + 30 x 7

= - 32 + 210 = 178

So, 31st term is 178