Exercise 31 Page 447

The explicit rule of an arithmetic sequence combines the information provided by the two equations of the recursive rule into a single equation. In both of these rules, $d$ is the common difference and $a_1$ is the first term. Let's start by substituting $1$ for $n$ in the given rule in order to find the value of $a_1.$ We know that $a_1=2,$ so now we need to identify the value of $d.$ Let's take a look at the given rule. Notice that the coefficient of $n$ is $\text{-}10,$ so we know that the terms are decreasing by $10$ each time. For the $n^{\text{th}}$ term, the number of times it has decreased in relation to $a_1$ is given by the value of $n.$ Therefore, the common difference $d$ is $\text{-}10.$ Now we have enough information to form a recursive rule for our sequence. Therefore, the recursive rule for our sequence is $a_1=2,$ $a_n=a_{n-1}-10.$