College Algebra (10th Edition)

$\bf\text{RECALL:}$ $\bf\text{(1) Arithmetic Sequence }$ 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 $\bf\text{Identify the sequence as arithmetic or geometric.}$ The sequence has $a_1$. $a_2=3$, $a_3=6$. $a_2-a_1 = 3-1=2 \\a_3-a_2=6-3=3$ There is no common difference, so the sequence is not arithmetic. $\dfrac{a_2}{a_1} = \dfrac{3}{1}=3 \\\dfrac{a_3}{a_2} = \dfrac{6}{2}=2$ There is no common ratio, so the sequence is not geometric.