Question

select the sequences that are geometric.
18, 36, 54, 72, ...
4.1, 8.2, 16.4, 32.8, ...
-7, 14, -28, 56, ...
980, 784, 627.2, 501.76, ...
5, 2, -1, -4, ...

which statements describe characteristics of a geometric sequence? check all that apply.
there is a common difference between terms
each term is multiplied by the same number to arrive at the next term.
the sequence increases or decreases in a linear pattern
there is a common ratio between terms.

Explanation:

Identify geometric sequences

Using the Geometric Sequences and Common Ratio knowledge points

• \(18, 36, 54, 72, \dots\):

$$ \frac{36}{18} = 2,\quad \frac{54}{36} = 1.5 \implies \text{Not geometric (Arithmetic with } d = 18) $$

• \(4.1, 8.2, 16.4, 32.8, \dots\):

$$ \frac{8.2}{4.1} = 2,\quad \frac{16.4}{8.2} = 2,\quad \frac{32.8}{16.4} = 2 \implies \text{Geometric with } r = 2 $$

• \(-7, 14, -28, 56, \dots\):

$$ \frac{14}{-7} = -2, \quad \frac{-28}{14} = -2, \quad \frac{56}{-28} = -2 \implies \text{Geometric with } r = -2 $$

• \(980, 784, 627.2, 501.76, \dots\):

$$ \frac{784}{980} = 0.8, \quad \frac{627.2}{784} = 0.8, \quad \frac{501.76}{627.2} = 0.8 \implies \text{Geometric with } r = 0.8 $$

• \(5, 2, -1, -4, \dots\):

$$ \frac{2}{5} = 0.4, \quad \frac{-1}{2} = -0.5 \implies \text{Not geometric (Arithmetic with } d = -3) $$

Analyze geometric sequence characteristics

Using the Geometric Sequences and Common Ratio knowledge points

• Statement 1: "There is a common difference between terms" \(\implies\) Describes arithmetic sequences, not geometric.
• Statement 2: "Each term is multiplied by the same number to arrive at the next term" \(\implies\) Correct definition of geometric progression.
• Statement 3: "The sequence increases or decreases in a linear pattern" \(\implies\) Describes arithmetic sequences.
• Statement 4: "There is a common ratio between terms" \(\implies\) Correct characteristic of geometric sequences.

Answer:

Question 1

The geometric sequences are:

• \(4.1, 8.2, 16.4, 32.8, \dots\)
• \(-7, 14, -28, 56, \dots\)
• \(980, 784, 627.2, 501.76, \dots\)

Question 2

The statements that describe characteristics of a geometric sequence are:

• Each term is multiplied by the same number to arrive at the next term.
• There is a common ratio between terms.