Question

What is the explicit formula for this geometric sequence?

64, 16, 4, 1,…

Answer 1

To find the explicit formula for the geometric sequence \(64, 16, 4, 1, \ldots\), we start by identifying the first term and the common ratio.

1. **Identify the first term (\(a\))**:
The first term \(a\) is \(64\).

2. **Determine the common ratio (\(r\))**:
The common ratio \(r\) can be found by dividing the second term by the first term:
\[
r = \frac{16}{64} = \frac{1}{4}
\]
We can confirm this by checking the ratio of other consecutive terms:
\[
\frac{4}{16} = \frac{1}{4}, \quad \frac{1}{4} = \frac{1}{4}
\]
So, the common ratio is indeed \(r = \frac{1}{4}\).

3. **Write the explicit formula**:
The explicit formula for a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Substituting \(a = 64\) and \(r = \frac{1}{4}\) into the formula, we obtain:
\[
a_n = 64 \cdot \left(\frac{1}{4}\right)^{n-1}
\]

Therefore, the explicit formula for the geometric sequence is:
\[
\boxed{64 \cdot \left(\frac{1}{4}\right)^{n-1}}
\]