Question

the sequences below are either arithmetic sequences or geometric sequences. for each sequer formula for the $n^{\text{th}}$ term $a_n$ of that sequence.
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$$\begin{tabular}{|c|c|c|} \\hline sequence & type & $n^{\\text{th}}$ term formula \\\\ \\hline (a) \\ 9, 13, 17, ... & $\\circ$ arithmetic \\\\ $\\circ$ geometric & $a_n = \\square$ \\\\ \\hline (b) \\ 6, 18, 54, ... & $\\circ$ arithmetic \\\\ $\\circ$ geometric & $a_n = \\square$ \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Analyze Sequence (a)

Check if it's arithmetic: Find the common difference. $13 - 9 = 4$, $17 - 13 = 4$. So it's arithmetic. The first term $a_1 = 9$, common difference $d = 4$. The formula for the $n$th term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$. Substitute values: $a_n = 9 + (n - 1) \times 4 = 9 + 4n - 4 = 4n + 5$.

Step2: Analyze Sequence (b)

Check if it's geometric: Find the common ratio. $\frac{18}{6} = 3$, $\frac{54}{18} = 3$. So it's geometric. The first term $a_1 = 6$, common ratio $r = 3$. The formula for the $n$th term of a geometric sequence is $a_n = a_1 \times r^{n - 1}$. Substitute values: $a_n = 6 \times 3^{n - 1}$.

Answer:

Part (a)

• Type: Arithmetic
• $a_n = 4n + 5$

Part (b)

• Type: Geometric
• $a_n = 6 \times 3^{n - 1}$