O Sequences, Series and Probabily Acthmetic and geometric sequences identifying and writing an explicit. The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric, and write the formula for the $n^{\text {th }}$ term $a_{n}$ of that sequence. \begin{tabular}{|c|c|c|} \hline Sequence & Type & $n^{\text {th }}$ term formula \\ \hline (a) $6,18,54, \ldots$ & \begin{tabular}{r} Arithmetic \\ Geometric \end{tabular} & $a_{n}=\square$ \\ \hline (b) $7,9,11, \ldots$ & \begin{tabular}{r} Arithmetic \\ Geometric \end{tabular} & $a_{n}=\square$ \\ \hline \end{tabular} \begin{tabular}{ccc} $\square^{\square}$ & $\frac{\square}{}$ & $\square$ 믐 \\ $\square+\square$ & $\square-\square$ & $\square \cdot \square$ \\ $x$ & & 6 \end{tabular}

Step 1 ：Identify whether each sequence is arithmetic or geometric. An arithmetic sequence is one in which the difference between any two successive members is a constant, while a geometric sequence is one in which each term after the first is found by multiplying the previous term by a fixed, non-zero number.

Step 2 ：For sequence (a) $6,18,54, \ldots$, each term is multiplied by 3 to get the next term, so it is a geometric sequence. The formula for the $n^{\text {th }}$ term $a_{n}$ of a geometric sequence is $a_{n}=a_{1} \cdot r^{(n-1)}$, where $a_{1}$ is the first term and $r$ is the common ratio.

Step 3 ：For sequence (b) $7,9,11, \ldots$, each term is increased by 2 to get the next term, so it is an arithmetic sequence. The formula for the $n^{\text {th }}$ term $a_{n}$ of an arithmetic sequence is $a_{n}=a_{1} + (n-1) \cdot d$, where $a_{1}$ is the first term and $d$ is the common difference.

Step 4 ：Calculate the $n^{\text {th }}$ term formula for each sequence. For sequence (a), $a_{n}=6 \cdot 3^{(n-1)}$. For sequence (b), $a_{n}=2n + 5$.

Step 5 ：The final answer is: \n\n\begin{tabular}{|c|c|c|}\n\hline Sequence & Type & $n^{\text {th }}$ term formula \\\n\hline (a) $6,18,54, \ldots$ & Geometric & $a_{n}=6 \cdot 3^{(n-1)}$ \\\n\hline (b) $7,9,11, \ldots$ & Arithmetic & $a_{n}=2n + 5$ \\\n\hline\n\end{tabular}