Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

4 , 16 , 64 , ...

Write your answer using decimals and integers.

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

1 answer

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:

\[ a_n = 4^n \]