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:
\[ a_n = 4^n \]
3 answers