Exercise 22 Page 447

We want to write a recursive rule for the given sequence. To do that, we first need to identify whether the given sequence is arithmetic, geometric, or neither. To do so we will calculate the difference and ratio between consecutive terms.

We can see above that neither the ratios nor the differences are the same. Therefore, the sequence is neither geometric nor arithmetic, so we have to look for another pattern. To do so we need to consider how the consecutive terms are related. Let's take a look at the differences between pairs of consecutive terms.

$a_n-a_{n-1}=n$

We can see above that the difference between the consecutive terms equals $n.$ Therefore, to obtain the value of the term in the $n^{\text{th}}$ position, we need to add $n$ to the previous term $a_{n-1}.$ With this information and knowing that the first term equals $\text{-}3$ we can write the recursive formula.