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# Find the 31st term of an AP whose $$11^{th}$$ term is 38 and $$16^{th}$$ term is 73.

Given $$a_{11} = 38$$ and $$a_{16} = 73$$

Using formula $$a_n = a + ( n - 1)d$$, to find nth term of arithmetic progression,

$$\Rightarrow$$ 38 = a + (11 - 1)(d), and
73 = a + (16 - 1)(d)
$$\Rightarrow$$ 38 = a + 10d, and
73 = a + 15d

These are equations consisting of two variables.

We have,
$$\Rightarrow$$ 38 = a + 10d
$$\Rightarrow$$ a = 38 - 10d
Let us put value of a in equation
$$\Rightarrow$$ (73 = a + 15d),
$$\Rightarrow$$ 73 = 38 - 10d + 15d
$$\Rightarrow$$ 35 = 5d

Therefore, Common difference = d = 7

Putting value of d in equation
$$\Rightarrow$$ 38 = a + 10d,
$$\Rightarrow$$ 38 = a + 70
$$\Rightarrow$$ a = -32

Therefore, common difference = d = 7 and
First term = a = –32

Using formula $$a_n = a + ( n - 1)d$$, to find nth term of arithmetic progression,

$$\Rightarrow a_{31}$$= -32 + (31 - 1) (7)
$$\Rightarrow$$ -32 + 210 = 178

Therefore, 31st term of AP is 178.