Question

Which formula can be used to find the nth term of the geometric sequence below?

1/6, 1, 6, 36...

Answer 1

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: \( \frac{1}{6}, 1, 6, 36 \)

1. The first term \( a_1 = \frac{1}{6} \).
2. 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 \( a_n = 6^{n-2} \).