Question

The first four terms of a sequence are shown.



one-eighth comma 1 comma 8 comma 64 comma ellipsis

Question
Which expression can be used to find the nth term in the sequence?

Answer options with 4 options
A.
one-eighth left-parenthesis eight right-parenthesis Superscript n minus 1 Baseline

B.
8 left-parenthesis one-eighth right-parenthesis Superscript n minus 1 Baseline

C.
one-eighth left-parenthesis 8 right-parenthesis Superscript n Baseline

D.
8 left-parenthesis one-eighth right-parenthesis Superscript n Baseline

Answer 1

To identify the nth term of the sequence: \(\frac{1}{8}, 1, 8, 64, \ldots\)

Let’s analyze the given terms:
1. First term (\(n=1\)): \(\frac{1}{8} = 8^{-1}\)
2. Second term (\(n=2\)): \(1 = 8^{0}\)
3. Third term (\(n=3\)): \(8 = 8^{1}\)
4. Fourth term (\(n=4\)): \(64 = 8^2\)

From this pattern, we can see that the nth term can be expressed in terms of powers of 8. Specifically, the nth term appears to be:

\[
8^{n-2}
\]

This means:
- For \(n=1\): \(8^{-1} = \frac{1}{8}\)
- For \(n=2\): \(8^{0} = 1\)
- For \(n=3\): \(8^{1} = 8\)
- For \(n=4\): \(8^{2} = 64\)

We can rewrite this expression in another form:
\[
8^{n-2} = \frac{1}{8}(8^{n-1})
\]

This form indicates that the nth term can also be defined as:
\[
\frac{1}{8} \cdot (8)^{n-1}
\]

Looking at the options:
- **Option A**: \(\frac{1}{8} (8)^{n-1}\) — this matches our derived expression.
- **Option B**: \(8 \cdot (\frac{1}{8})^{n-1}\) — incorrect as it gives outputs not in line with the pattern.
- **Option C**: \(\frac{1}{8} (8)^{n}\) — incorrect because it shifts the exponent and produces different values.
- **Option D**: \(8 \cdot (\frac{1}{8})^{n}\) — incorrect for the same reason as B.

The correct expression to find the nth term in the sequence is:

**A. \(\frac{1}{8} (8)^{n-1}\)**