Answer
Neither arithmetic nor geometric
Work Step by Step
We are given the sequence:
$\{5n^2+1\}$
Compute the difference between two consecutive terms:
$a_{n+1}-a_n=(5(n+1)^2+1)-(5n^2+1)=5(n^2+2n+1)+1-5n^2-1=5n^2+10n+5-5n^2-1=10n+5$
As the difference between any consecutive terms is not constant, the sequence is not arithmetic.
Compute the ratio between two consecutive terms:
$\dfrac{a_{n+1}}{a_n}=\dfrac{5(n+1)^2+1}{5n^2+1}=\dfrac{5n^2+10n+6}{5n^2+1}$
As the ratio between any consecutive terms is not constant, the sequence is not geometric.
So the sequence is neither arithmetic nor geometric.