Question

What is the explicit formula for this geometric sequence?
27, 9, 3, 1, ...

Answer 1

To find the explicit formula for the given geometric sequence \(27, 9, 3, 1, \ldots\), we first identify the first term and the common ratio.

1. The first term \(a\) is 27.
2. The common ratio \(r\) can be found by dividing the second term by the first term:
\[
r = \frac{9}{27} = \frac{1}{3}.
\]

A geometric sequence can be expressed in the form:
\[
a_n = a \cdot r^{n-1},
\]
where:
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number (with \(n=1\) for the first term).

Substituting the values we know:
- \(a = 27\),
- \(r = \frac{1}{3}\),

we get the explicit formula:
\[
a_n = 27 \cdot \left(\frac{1}{3}\right)^{n-1}.
\]

Thus, the explicit formula for the geometric sequence is:
\[
a_n = 27 \cdot \left(\frac{1}{3}\right)^{n-1}.
\]