Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

4 , 16 , 64 , ...


Write your answer using decimals and integers.

an = _(_)^ n^ -^ 1

Answer 1

To find the equation that describes the given sequence \(4, 16, 64, \ldots\), we first need to understand the pattern in the sequence.

We can express each term as follows:

- The first term (\(n = 1\)) is \(4\).
- The second term (\(n = 2\)) is \(16\). This can be expressed as \(4^2\).
- The third term (\(n = 3\)) is \(64\). This can be expressed as \(4^3\).

From this, we can see that each term can be represented as \(4^n\) where \(n\) is the position of the term.

Therefore, the equation to describe the sequence is:

\[
a_n = 4^n
\]

If we want to use the form you indicated with a modification, we can express it as:

\[
a_n = 4^{n} - 1
\]

This is not a direct expression of your sequence; it only describes the power of \(4\). Hence, the most straightforward representation of the sequence you're looking for is simply:

\[
a_n = 4^n
\]

If you specifically mean to follow your format, it could be interpreted as:

\[
a_n = 4^{n} - 1 = (2^2)^{n} - 1
\]

However, that would be misleading as it would not yield the original numbers in the sequence before the subtraction. Thus, the simplest form remains:

\[
a_n = 4^n
\]