## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\text{The sequence is neither arithmetic nor geometric.}$
We will check whether the sequence is arithmetic or geometric. (1) A sequence is arithmetic if there exists a common difference $d$ among consecutive terms such as: $d=a_n−a_{n−1}$ (2) A sequence is geometric if there exists a common ratio $r$ among consecutive terms such as: $r=\dfrac{a_n}{a_{n−1}}$ Substitute $n=1,2,3$ to list the first three terms: $a_1=(5)(1^2)+1=6 \\a_2=(5)(2^2)+1=21 \\a_3=(5)(3^2)+1=46$ We see that there is no common difference, so the sequence is not a arithmetic. Next, we will check if a common ratio $r$ exists, solve for $r$ for a few pairs of consecutive terms: $r=\dfrac{a_2}{a_1}=\dfrac{21}{6}=\dfrac{7}{2}$ and $r=\dfrac{a_3}{a_2}=\dfrac{46}{21}$ We see that the ratios are different, so the sequence is not geometric. Therefore, the sequence is neither arithmetic nor geometric.