determine if the sequence is arithmetic or geometric and determine the common difference / ratio in simplest form 147,21,3

To determine if the sequence \(147, 21, 3\) is arithmetic or geometric, let's first explore both possibilities:

### 1. Arithmetic Sequence:
In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term.

- The difference between the second and first terms: 
  \[
  21 - 147 = -126
  \]
- The difference between the third and second terms: 
  \[
  3 - 21 = -18
  \]

Since \(-126 \neq -18\), this sequence is **not arithmetic**.

### 2. Geometric Sequence:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.

- The ratio between the second and first terms: 
  \[
  \frac{21}{147} = \frac{1}{7}
  \]
- The ratio between the third and second terms: 
  \[
  \frac{3}{21} = \frac{1}{7}
  \]

Since both ratios are \(\frac{1}{7}\), the sequence is **geometric**, with a common ratio of \(\frac{1}{7}\).

### Conclusion:
- The sequence is **geometric**.
- The common ratio is \(\frac{1}{7}\).

Do you want more details or have any questions? 

Here are 5 related questions for further exploration:
1. How do you find the nth term of a geometric sequence?
2. What are the properties of geometric sequences?
3. Can a geometric sequence have negative terms?
4. How would the sequence change if the common ratio were negative?
5. What happens when the common ratio is greater than 1 in a geometric sequence?

**Tip:** In geometric sequences, when the common ratio is a fraction, each term becomes smaller than the previous one.

Determine if the sequence is arithmetic or geometric and determine the common difference / ratio in simplest form: 147, 21, 3.

Learn how to determine if the sequence 147, 21, 3 is arithmetic or geometric. The solution explores both types of sequences and shows that the sequence is geometric with a common ratio of 1/7. Suitable for students in Grades 7-9.

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