https://file.aiquickdraw.com/mathbot/file/1725253337240bpnox88g.jpg

The image provided shows a sequence of numbers, and the question asks whether the sequence is arithmetic or geometric. The sequence in the image is:

\[ 2, 4, 8, 16, 32, 64 \]

To determine whether the sequence is arithmetic or geometric:

1. **Arithmetic Sequence**: A sequence is arithmetic if the difference between consecutive terms is constant. In other words, \(a_{n+1} - a_n = d\) where \(d\) is a constant.

2. **Geometric Sequence**: A sequence is geometric if the ratio between consecutive terms is constant. In other words, \(\frac{a_{n+1}}{a_n} = r\) where \(r\) is a constant.

### Checking the Sequence:
- The difference between consecutive terms:
  - \(4 - 2 = 2\)
  - \(8 - 4 = 4\)
  - \(16 - 8 = 8\)
  - \(32 - 16 = 16\)
  - \(64 - 32 = 32\)
  
  Since the differences are not constant, the sequence is **not arithmetic**.

- The ratio between consecutive terms:
  - \(\frac{4}{2} = 2\)
  - \(\frac{8}{4} = 2\)
  - \(\frac{16}{8} = 2\)
  - \(\frac{32}{16} = 2\)
  - \(\frac{64}{32} = 2\)

  Since the ratio is constant, the sequence is **geometric** with a common ratio \(r = 2\).

**Conclusion**: The sequence is **geometric**.

Do you want more details or have any other questions?

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Here are 5 related questions:
1. How can you determine the nth term of a geometric sequence?
2. What is the formula for the sum of the first n terms of a geometric series?
3. Can a sequence be both arithmetic and geometric? Why or why not?
4. What is the difference between finite and infinite geometric sequences?
5. How does the common ratio affect the behavior of a geometric sequence?

**Tip**: Always check the difference and ratio between terms when determining if a sequence is arithmetic or geometric.

Explore whether a sequence is arithmetic or geometric with detailed step-by-step analysis. Understand the difference and ratio between terms to determine the nature of the sequence. Suitable for high school students studying sequences and series in mathematics.

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