https://file.aiquickdraw.com/mathbot/file/1725863148148ul0g93f6.jpg

Let's analyze and solve the problem step by step:

### Part (a): Determine whether the sequence is arithmetic, geometric, or neither.

The sequence is:  
60, 30, 15, \( \frac{15}{2} \), ...

- **Arithmetic Sequence**: A sequence where each term is obtained by adding a constant difference. 
  Let's check the difference between consecutive terms:
  \[
  30 - 60 = -30
  \]
  \[
  15 - 30 = -15
  \]
  \[
  \frac{15}{2} - 15 = \frac{15}{2} - \frac{30}{2} = -\frac{15}{2}
  \]
  Since the differences are not the same, the sequence is **not arithmetic**.

- **Geometric Sequence**: A sequence where each term is obtained by multiplying the previous term by a constant ratio.
  Let's check the ratio between consecutive terms:
  \[
  \frac{30}{60} = \frac{1}{2}, \quad \frac{15}{30} = \frac{1}{2}, \quad \frac{\frac{15}{2}}{15} = \frac{1}{2}
  \]
  Since the ratio between each pair of consecutive terms is \( \frac{1}{2} \), the sequence is **geometric**.

### Part (b): Recursive equation.

A recursive equation defines each term in the sequence based on the previous term. 

Since this is a geometric sequence with a common ratio of \( \frac{1}{2} \), the recursive equation is:
\[
a_{n+1} = \frac{1}{2} a_n \quad \text{with} \quad a_1 = 60
\]

### Part (c): Explicit equation.

For a geometric sequence, the explicit formula is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \( a_1 \) is the first term,
- \( r \) is the common ratio, and
- \( n \) is the term number.

Here, \( a_1 = 60 \) and \( r = \frac{1}{2} \). Thus, the explicit formula is:
\[
a_n = 60 \cdot \left( \frac{1}{2} \right)^{n-1}
\]

### Summary:
- (a) The sequence is **geometric**.
- (b) The recursive equation is \( a_{n+1} = \frac{1}{2} a_n \) with \( a_1 = 60 \).
- (c) The explicit equation is \( a_n = 60 \cdot \left( \frac{1}{2} \right)^{n-1} \).

Let me know if you want further details, or if anything is unclear! Here are 5 related questions:

1. How can we identify if a sequence is arithmetic or geometric?
2. What are the differences between explicit and recursive formulas?
3. How can the common ratio of a geometric sequence be determined from any two terms?
4. What happens to the sequence if the common ratio is negative?
5. How can we use the explicit formula to find the 10th term in this sequence?

**Tip:** When solving for sequences, always check for common differences (arithmetic) or ratios (geometric) to quickly classify the sequence type!

60, 30, 15, 15/2, ...
a. Arithmetic, geometric, or neither?
b. Recursive equation:
c. Explicit equation:

Solve the geometric sequence problem with terms 60, 30, 15, and 15/2. Learn how to identify the sequence type, write recursive and explicit formulas, and understand the core concepts of geometric sequences. Suitable for high school students in Grades 9-12.

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