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

1. Check if it's Arithmetic:

For a sequence to be arithmetic, the difference between consecutive terms must be constant.

The first few differences are:

• $-3 - 0.3 = -3.3$
• $30 - (-3) = 33$
• $-300 - 30 = -330$

Clearly, the differences are not constant, so the sequence is not arithmetic.

2. Check if it's Geometric:

For a sequence to be geometric, the ratio between consecutive terms must be constant.

The ratios between consecutive terms are:

• Ratio between second and first term:
  $r_1 = \frac{-3}{0.3} = -10$
• Ratio between third and second term:
  $r_2 = \frac{30}{-3} = -10$
• Ratio between fourth and third term:
  $r_3 = \frac{-300}{30} = -10$
• Ratio between fifth and fourth term:
  $r_4 = \frac{3000}{-300} = -10$

The ratio is consistently -10, so the sequence is geometric with a common ratio of $r = -10$.

Final Answer:

• The sequence is geometric.
• The common ratio is $\mathbf{-10}$.

Do you want a detailed explanation on geometric sequences or have any further questions?

Related Questions:

1. What is the formula for the $n$-th term of a geometric sequence?
2. How do you find the sum of a finite geometric sequence?
3. How does a negative common ratio affect the sequence?
4. What is the difference between arithmetic and geometric sequences?
5. Can a sequence be both arithmetic and geometric?

Tip:

In a geometric sequence, to confirm the common ratio $r$, divide any term by its preceding term. If $r$ is constant throughout, the sequence is geometric.