Let's analyze the given sequences and determine if they are arithmetic or geometric, and then find their $n$-th term formulas:

Sequence (a): $6, 18, 54, \dots$

• Type: To check if the sequence is geometric, divide consecutive terms: $\frac{18}{6} = 3, \quad \frac{54}{18} = 3$ The common ratio ($r$) is $3$. Thus, this is a geometric sequence.

• Formula: The formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Here, $a_1 = 6$ and $r = 3$. Therefore: $a_n = 6 \cdot 3^{n-1}$

Sequence (b): $18, 22, 26, \dots$

• Type: To check if the sequence is arithmetic, subtract consecutive terms: $22 - 18 = 4, \quad 26 - 22 = 4$ The common difference ($d$) is $4$. Thus, this is an arithmetic sequence.

• Formula: The formula for an arithmetic sequence is: $a_n = a_1 + (n-1)d$ Here, $a_1 = 18$ and $d = 4$. Therefore: $a_n = 18 + (n-1) \cdot 4$ Simplify: $a_n = 4n + 14$

Final Answer:

1. Sequence (a):

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

2. Sequence (b):

   • Type: Arithmetic
   • Formula: $a_n = 4n + 14$

Would you like further clarification or details on these formulas?

Here are five related questions to expand your understanding:

1. How do you generalize the formula for a geometric sequence if the common ratio is negative?
2. What happens to an arithmetic sequence when the common difference is $0$?
3. How can you identify whether a sequence is neither arithmetic nor geometric?
4. Can a sequence be both arithmetic and geometric simultaneously? Why or why not?
5. What is the sum of the first $n$ terms for an arithmetic sequence?

Tip: Always verify a few terms of the sequence using your formula to ensure correctness!