To determine whether the sequence $0.3, -3, 30, -300, 3000, \dots$ is arithmetic, geometric, or neither, let us analyze the pattern step by step.

Step 1: Check if the sequence is arithmetic.

In an arithmetic sequence, the difference between consecutive terms (common difference) remains constant.

• First difference: $-3 - 0.3 = -3.3$
• Second difference: $30 - (-3) = 33$

Since the differences are not constant, the sequence is not arithmetic.

Step 2: Check if the sequence is geometric.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio $r$.

To find $r$, divide consecutive terms:

1. $r_1 = \frac{-3}{0.3} = -10$
2. $r_2 = \frac{30}{-3} = -10$
3. $r_3 = \frac{-300}{30} = -10$
4. $r_4 = \frac{3000}{-300} = -10$

The ratio $r = -10$ is consistent between terms, confirming that the sequence is geometric.

Step 3: State the common ratio.

The common ratio is:

$r = -10$

Final Answer:

• The sequence is geometric.
• The common ratio is $-10$.

Would you like me to explain geometric sequences further or verify another property? Let me know!

Related Questions:

1. What is the formula for the $n$-th term of a geometric sequence?
2. How do you determine the sum of a geometric sequence?
3. What happens to a geometric sequence when the common ratio is negative?
4. What is the difference between arithmetic and geometric sequences?
5. How do you solve problems involving infinite geometric series?

Tip:

In a geometric sequence, the ratio $r$ can be positive or negative, but it must remain constant for all consecutive terms.