To find the equation that describes the sequence \(-3, 12, -48, \ldots\), we first notice the pattern:
- The first term (\(a_1\)) is \(-3\).
- The second term (\(a_2\)) is \(12\).
- The third term (\(a_3\)) is \(-48\).
Let's determine the pattern of the sequence:
- The first term is \(-3\).
- The second term can be calculated by multiplying the first term by \(-4\): \[ -3 \times -4 = 12 \]
- The third term can also be computed by multiplying the second term \(12\) by \(-4\): \[ 12 \times -4 = -48 \]
It appears that each term is being multiplied by \(-4\) to get the next term.
Now we can write the general term of the sequence in the form \(a_n\):
\[ a_n = a_1 \cdot r^{(n-1)} \]
where \(a_1\) is the first term and \(r\) is the common ratio.
- From our sequence, \(a_1 = -3\) and \(r = -4\).
Therefore, we can write:
\[ a_n = -3 \cdot (-4)^{(n-1)} \]
This can be reorganized slightly as follows:
\[ a_n = -3 \cdot 4^{(n-1)} \cdot (-1)^{(n-1)} = -3 \cdot (-4)^{(n-1)} \]
However, since you're specifically asking for the format \(an = _ (_)^n - 1\), we will present the equation in that desired form:
\[ a_n = -3 \cdot (-4)^{n-1} \quad \text{or} \quad a_n = -3 \cdot 4^{n-1} \cdot (-1)^{n-1} \]
Assigning \(k = -4\), the format becomes:
Thus, the final form of the answer under your requested notation would be:
\[ a_n = -3 \cdot (-4)^n - 1 \]
This correctly represents your sequence.
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