To find the common ratio of the given sequence, let's analyze the terms provided:
1st term: [tex]\(\frac{2}{3}\)[/tex]
2nd term: [tex]\(\frac{1}{6}\)[/tex]
3rd term: [tex]\(\frac{1}{24}\)[/tex]
4th term: [tex]\(\frac{1}{96}\)[/tex]
The common ratio [tex]\(r\)[/tex] in a geometric sequence is found by dividing any term by the previous term. Let's look at the first and second terms to find the common ratio:
[tex]\[
r = \frac{\text{2nd term}}{\text{1st term}} = \frac{\frac{1}{6}}{\frac{2}{3}}
\][/tex]
Now, performing this division:
1. When you divide fractions, you multiply by the reciprocal of the denominator:
[tex]\[
r = \frac{1}{6} \times \frac{3}{2}
\][/tex]
2. Simplify the multiplication:
[tex]\[
r = \frac{1 \times 3}{6 \times 2} = \frac{3}{12} = \frac{1}{4}
\][/tex]
Therefore, the common ratio [tex]\(r\)[/tex] of the sequence is [tex]\(\frac{1}{4}\)[/tex].
Given the choices:
- [tex]\(\frac{1}{9}\)[/tex]
- [tex]\(\frac{1}{4}\)[/tex]
- 4
- 9
The correct answer is:
[tex]\(\frac{1}{4}\)[/tex].
