Question

vince wrote the sequence below.

\\(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \dots\\)

which of the following explains whether the sequence is geometric?

• the sequence is geometric because \\(\frac{2}{3}\\) was added to each term to get the next term.
• the sequence is geometric because 3 was multiplied to each term to get the next term.
• the sequence is not geometric because \\(\frac{2}{3}\\) was added to each term to get the next term.
• the sequence is not geometric because 3 was multiplied to each term to get the next term.

Explanation:

Analyze the given sequence

Using the Geometric Sequences knowledge point
The sequence is:

$$ \frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \dots $$

Rewrite \(1\) as a fraction with a denominator of \(3\):

$$ \frac{1}{3}, \frac{3}{3}, \frac{5}{3}, \frac{7}{3}, \dots $$

Test for a common ratio

Using the Common Ratio knowledge point
Calculate the ratios of consecutive terms:

$$
LATEXBLOCK0
$$

Since \(r_1
eq r_2\), there is no common ratio, so the sequence is not geometric.

Determine the pattern of the sequence

Find the difference between consecutive terms:

$$
LATEXBLOCK1
$$

The sequence is arithmetic because a constant value of \(\frac{2}{3}\) is added to each term to get the next term. Thus, it is not geometric.

Answer:

• The sequence is geometric because \(\frac{2}{3}\) was added to each term to get the next term.
• The sequence is geometric because 3 was multiplied to each term to get the next term.
• The sequence is not geometric because \(\frac{2}{3}\) was added to each term to get the next term. (Correct answer)
• The sequence is not geometric because 3 was multiplied to each term to get the next term.