Answer
The sequence is neither arithmetic nor geometric.
Work Step by Step
$\bf\text{RECALL:}$
$\bf\text{(1) Arithmetic Sequence }$
A sequence is arithmetic if there exists a common difference $d$ among consecutive terms.
$d=a_n-a_{n-1}$
The sum of the first $n$ terms of an arithmetic sequence is given by the formulas:
$S_n=\frac{n}{2}(a_1 +a_n)$
or
$S_n=\frac{n}{2}\left(2 a_1 + (n-1)d\right)$
$\bf\text{(2) Geometric Sequence }$
A sequence is geometric if there exists a common ratio $r$ among consecutive terms.
$r=\dfrac{a_n}{a_{n-1}}$
The sum of the first $n$ terms of a geometric sequence is given by the formula:
$S_{n}=a_1 \cdot \dfrac{1-r^n}{1-r}$
In the formulas listed above,
$d$ = common difference
$r$ = common ratio
$a_1$ = first term
$a_n$ = nth term
$n$ = number of terms
$\bf\text{List the first few terms of the sequence.}$
$\bf\text{Identify sequence as arithmetic or geometric.}$
Substitute $1, 2, 3$ for $n$ to list the first three terms:
$a_1 =5(1^2)+1=5(1)+1=6
\\a_2 = 5(2^2)+1=5(4)+1=21
\\a_3 = 5(3^2)+1=5(9)+1=46$
The sequence has no common difference.
To check if a common ratio exists, solve for $r$ for a few pairs of consecutive terms:
$r=\dfrac{a_2}{a_1} = \dfrac{21}{6} = \dfrac{7}{2}
\\r=\dfrac{a_3}{a_2} = \dfrac{46}{21}$
The ratios are different, so the sequence is not geometric.
