To determine whether the given sequence [tex]\(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots\)[/tex] is geometric or not, we need to understand the properties of geometric and arithmetic sequences.
A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a constant called the common ratio.
An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant called the common difference to the previous term.
Given sequence: [tex]\(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}\)[/tex]
1. Check if the sequence is geometric:
- Calculate the ratios between consecutive terms:
[tex]\[
\frac{1}{3} \text{ and } 1: \quad \frac{1}{1/3} = 3
\][/tex]
[tex]\[
1 \text{ and } \frac{5}{3}: \quad \frac{5/3}{1} = \frac{5}{3}
\][/tex]
[tex]\[
\frac{5}{3} \text{ and } \frac{7}{3}: \quad \frac{7/3}{5/3} = \frac{7}{5}
\][/tex]
- The ratios are [tex]\( \frac{3}{1}, \frac{5}{3}, \frac{7}{5} \)[/tex]. Since the ratios are not equal, the sequence is not geometric.
2. Check if the sequence is arithmetic:
- Calculate the differences between consecutive terms:
[tex]\[
1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\][/tex]
[tex]\[
\frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}
\][/tex]
[tex]\[
\frac{7}{3} - \frac{5}{3} = \frac{7}{3} - \frac{5}{3} = \frac{2}{3}
\][/tex]
- The differences are [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{2}{3}\)[/tex]. Since the differences are equal, the sequence is arithmetic.
Given the sequence is arithmetic but not geometric, we need to identify the correct explanation from the options provided:
- "The sequence is geometric because [tex]\(\frac{2}{3}\)[/tex] was added to each term to get the next" is incorrect because the sequence is not geometric and adding a constant is a property of arithmetic sequences.
- "The sequence is geometric because 3 was multiplied to each term to get the next" is incorrect because the common ratio is not consistent or equal to 3.
- "The sequence is not geometric because [tex]\(\frac{2}{3}\)[/tex] was added to each term to get the next" is correct because it correctly identifies the sequence as arithmetic and explains why it is not geometric.
- "The sequence is not geometric because 3 was multiplied to each term to get the next" is incorrect because the sequence is neither geometric with a multiplier of 3.
Thus, the correct explanation is:
"The sequence is not geometric because [tex]\(\frac{2}{3}\)[/tex] was added to each term to get the next."
So the correct option is:
"The sequence is not geometric because [tex]\(\frac{2}{3}\)[/tex] was added to each term to get the next term."
