To determine the common ratio of the sequence [tex]\(27, 9, 3, 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\)[/tex], follow these steps:
1. Identify Successive Terms:
Let's take the first two terms from the sequence. The first term ([tex]\(a_1\)[/tex]) is 27, and the second term ([tex]\(a_2\)[/tex]) is 9.
2. Calculate the Common Ratio [tex]\(r\)[/tex]:
The common ratio [tex]\(r\)[/tex] in a geometric sequence is calculated by dividing any term by its preceding term. Here, we will divide the second term by the first term:
[tex]\[
r = \frac{a_2}{a_1} = \frac{9}{27}
\][/tex]
3. Simplify the Fraction:
Simplify [tex]\(\frac{9}{27}\)[/tex] as follows:
[tex]\[
\frac{9}{27} = \frac{9 \div 9}{27 \div 9} = \frac{1}{3}
\][/tex]
4. Verify the Result:
For further confirmation, you can check the ratio for other successive terms. Taking the third term ([tex]\(a_3 = 3\)[/tex]) and dividing it by the second term ([tex]\(a_2 = 9\)[/tex]):
[tex]\[
r = \frac{3}{9} = \frac{1}{3}
\][/tex]
Similarly, dividing the fourth term ([tex]\(a_4 = 1\)[/tex]) by the third term ([tex]\(a_3 = 3\)[/tex]) also yields:
[tex]\[
r = \frac{1}{3}
\][/tex]
So, the common ratio [tex]\(r\)[/tex] between successive terms in the sequence is [tex]\(\frac{1}{3}\)[/tex].
Thus, the correct choice among the options is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
