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.