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

Let $a =$ first term, $d =$ common difference.

$$\begin{aligned} t_{11} &= a + 10d = 38 & \dots(i) \\ t_{16} &= a + 15d = 73 & \dots(ii) \end{aligned}$$

Subtract (i) from (ii):

$$5d = 35 \Rightarrow d = 7$$

From (i): $a = 38 - 70 = -32$.

$$t_{31} = a + 30d = -32 + 210 = 178$$

Therefore, the $31^{\text{st}}$ term is 178.