To find the nth term of a geometric sequence, we can use the formula:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where:
- \( a_n \) is the nth term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio, and
- \( n \) is the term number.
In your geometric sequence: \( \frac{1}{6}, 1, 6, 36 \)
- The first term \( a_1 = \frac{1}{6} \).
- To find the common ratio \( r \), divide the second term by the first term:
\[ r = \frac{1}{\frac{1}{6}} = 6 \]
To verify, divide the third term by the second term:
\[ \frac{6}{1} = 6 \]
And the fourth term by the third term:
\[ \frac{36}{6} = 6 \]
Since \( r = 6 \) is consistent, we confirm the common ratio is 6.
Now, substituting \( a_1 \) and \( r \) into the formula:
\[ a_n = \frac{1}{6} \cdot 6^{(n-1)} \]
Therefore, the formula for the nth term of the sequence is:
\[ a_n = \frac{1}{6} \cdot 6^{(n-1)} \]
This can also be expressed as:
\[ a_n = \frac{6^{n-1}}{6} = 6^{n-2} \]
So, the nth term can also be represented as \( a_n = 6^{n-2} \).