Question: Vince wrote the given sequence. \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots\) Which…

Vince wrote the given sequence.

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

Which of the following explains whether the sequence is geometric?

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

Solution

The sequence given is \( \frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots \). To determine if this sequence is geometric, we need to check if there is a constant ratio between consecutive terms. First, find the ratio between the first and second terms: \[ \frac{1}{1/3} = 3 \] The ratio is 3. Next, find the ratio between the second and third terms: \[ \frac{5/3}{1} = \frac{5}{3} \] Since the ratios are not consistent (3 and \(\frac{5}{3}\)), the sequence is not geometric. Therefore, the correct explanation is: “The sequence is not geometric because 3 was multiplied to each term to get the next term.”