The question asks to identify whether the given sequences are arithmetic, geometric, or neither, and, for arithmetic and geometric sequences, to determine the growth pattern. Let us analyze each sequence:

Sequence Analysis:

(a) 12, 144, 1728,...

• To check if it's arithmetic, calculate the difference between consecutive terms:
  $144 - 12 = 132$, $1728 - 144 = 1584$.
  Since the difference is not constant, it's not arithmetic.
• To check if it's geometric, calculate the ratio between consecutive terms:
  $\frac{144}{12} = 12$, $\frac{1728}{144} = 12$.
  Since the ratio is constant ($r = 12$), this is a geometric sequence with a growth pattern of multiplying by 12.

(b) 0, 5, 10, 15, 20, 25,...

• To check if it's arithmetic, calculate the difference between consecutive terms:
  $5 - 0 = 5$, $10 - 5 = 5$, $15 - 10 = 5$.
  Since the difference is constant ($d = 5$), this is an arithmetic sequence with a growth pattern of adding 5.
• It is not geometric because the ratio between terms ($\frac{5}{0}, \frac{10}{5}$) is not consistent.

(c) 0, 4, 16, 36, 64,...

• To check if it's arithmetic, calculate the differences:
  $4 - 0 = 4$, $16 - 4 = 12$, $36 - 16 = 20$.
  The differences are not constant, so it's not arithmetic.
• To check if it's geometric, calculate the ratios:
  $\frac{4}{0}, \frac{16}{4} = 4, \frac{36}{16} = 2.25$.
  The ratios are not constant, so it's not geometric either.
  This is neither arithmetic nor geometric.

(d) 1.5, 2.25, 3.375, 5.0625,...

• To check if it's arithmetic, calculate the differences:
  $2.25 - 1.5 = 0.75$, $3.375 - 2.25 = 1.125$.
  The differences are not constant, so it's not arithmetic.
• To check if it's geometric, calculate the ratios:
  $\frac{2.25}{1.5} = 1.5$, $\frac{3.375}{2.25} = 1.5$.
  Since the ratio is constant ($r = 1.5$), this is a geometric sequence with a growth pattern of multiplying by 1.5.

Summary of Results:

• (a): Geometric, growth pattern = multiply by 12.
• (b): Arithmetic, growth pattern = add 5.
• (c): Neither arithmetic nor geometric.
• (d): Geometric, growth pattern = multiply by 1.5.

Would you like me to explain any of these in greater detail?

Follow-up Questions:

1. What are the formulas for the $n$-th term of an arithmetic or geometric sequence?
2. How can we classify a sequence as "neither arithmetic nor geometric"?
3. Can a sequence be both arithmetic and geometric simultaneously? If so, under what condition?
4. How do we calculate the sum of the first $n$ terms of an arithmetic sequence?
5. How do we calculate the sum of the first $n$ terms of a geometric sequence?

Tip:

For any sequence, always check the first few terms to determine if there's a common difference (arithmetic) or ratio (geometric).