Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm

–3, 12,–48,...
Write your answer using decimals and integers.
an = _(_)^n^ -^ 1

Answer 1

To find the equation that describes the sequence \(-3, 12, -48, \ldots\), we first notice the pattern:

1. The first term (\(a_1\)) is \(-3\).
2. The second term (\(a_2\)) is \(12\).
3. 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.