Answer
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 that: $d=a_n−a_{n−1}$
The sum of the first n terms of an arithmetic sequence can be computed as:
$S_n=\dfrac{n}{2}[2a_1+(n−1)d] (1)$
(2) A sequence is geometric if there exists a common ratio $r$ among consecutive terms such that: $r=\dfrac{a_n}{a_{n−1}}$
The sum of the first n terms of a geometric sequence can be computed as:
$S_n= \dfrac{a_1(1-r^n)}{1-r} (2)$
where, $a_1 =\ First \ term$, $a_n$ = $n$th term, and $n =\ Number \ of \ Terms$
We have: $a_2-a_1=3-1 =2 \\ a_3-a_2=6-3=3$
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:
$r=\dfrac{a_2}{a_1}=\dfrac{3}{1}=3$ and $ r=\dfrac{a_3}{a_2}=\dfrac{6}{3}=2$
We see that the ratios are different, so the sequence is not geometric.
Therefore, the sequence is neither arithmetic nor geometric.
