Answer
$\text{The sequence is neither arithmetic nor geometric.}$
Work Step by Step
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.
