To find the nth term of the geometric sequence given by [tex]\(\frac{1}{6}, 1, 6, 36, \ldots\)[/tex], we need to figure out the initial term and the common ratio of the sequence. Let's go through this step-by-step.
1. Identify the first term ([tex]\(a\)[/tex]) of the sequence:
The first term of the sequence is [tex]\(\frac{1}{6}\)[/tex].
2. Determine the common ratio ([tex]\(r\)[/tex]) of the sequence:
To find the common ratio, we divide any term in the sequence by the previous term. Let's take the second term and divide it by the first term:
[tex]\[
r = \frac{1}{\frac{1}{6}} = \frac{1 \times 6}{1} = 6
\][/tex]
3. Verify the common ratio with subsequent terms:
To ensure that the common ratio is consistent, let's check it with the next term. The third term divided by the second term should also give us the common ratio:
[tex]\[
r = \frac{6}{1} = 6
\][/tex]
Similarly, the fourth term divided by the third term should also be:
[tex]\[
r = \frac{36}{6} = 6
\][/tex]
We have confirmed that the common ratio is indeed 6 throughout the sequence.
4. Write the formula for the nth term ([tex]\(a_n\)[/tex]) of a geometric sequence:
The general formula for the nth term of a geometric sequence is given by:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
Here, [tex]\(a\)[/tex] is the first term, and [tex]\(r\)[/tex] is the common ratio.
5. Substitute the values of the initial term and the common ratio into the formula:
We have [tex]\(a = \frac{1}{6}\)[/tex] and [tex]\(r = 6\)[/tex]. Substituting these into the formula gives us:
[tex]\[
a_n = \frac{1}{6} \cdot 6^{n-1}
\][/tex]
Thus, the formula to find the nth term of the given geometric sequence is:
[tex]\[
a_n = \frac{1}{6} \cdot 6^{n-1}
\][/tex]
So, the correct option from the given choices is:
[tex]\[
\boxed{a_n = \frac{1}{6} \cdot 6^{n-1}}
\][/tex]
