Let’s determine the common ratio for the given sequence:
[tex]\[ 27, 9, 3, 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots \][/tex]
In a geometric sequence, the common ratio [tex]\( r \)[/tex] is the factor by which we multiply each term to get the next term. We can find this by dividing any term by the preceding term.
1. Finding the common ratio step-by-step:
- Divide the second term by the first term:
[tex]\[
\frac{9}{27} = \frac{1}{3}
\][/tex]
- Divide the third term by the second term:
[tex]\[
\frac{3}{9} = \frac{1}{3}
\][/tex]
- Divide the fourth term by the third term:
[tex]\[
\frac{1}{3} = \frac{1}{3}
\][/tex]
- Divide the fifth term by the fourth term:
[tex]\[
\left( \frac{1}{3} \right) \div 1 = \frac{1}{3}
\][/tex]
- Divide the sixth term by the fifth term:
[tex]\[
\left( \frac{1}{9} \right) \div \left( \frac{1}{3} \right) = \frac{1}{3}
\][/tex]
- Divide the seventh term by the sixth term:
[tex]\[
\left( \frac{1}{27} \right) \div \left( \frac{1}{9} \right) = \frac{1}{3}
\][/tex]
2. Conclusion:
Each division yields the same result:
[tex]\[
\frac{1}{3}
\][/tex]
Thus, the common ratio [tex]\( r \)[/tex] for the given sequence is:
[tex]\[ \frac{1}{3} \][/tex]
This matches the options given in the choices: [tex]\(\frac{1}{3}\)[/tex].
