Question

Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term.
-1, 3, -9, ...
Write your answer using decimals and integers.
an = ____ (___)^n-1

Answer 1

To find the equation for the given sequence \(-1, 3, -9, ...\), we first identify the pattern in the terms:

1. The first term \(a_1 = -1\)
2. The second term \(a_2 = 3\)
3. The third term \(a_3 = -9\)

Next, let's express each term in terms of a common base and a factor that varies based on \(n\):

We see that the terms alternate in sign, and the absolute values of the terms appear to be multiplying by \(-3\) for each term:
- From \(-1\) to \(3\): \(-1 \times -3 = 3\)
- From \(3\) to \(-9\): \(3 \times -3 = -9\)

This suggests a pattern where each term is obtained by multiplying the previous term by \(-3\).

The general form of a geometric sequence is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
where \(a_1\) is the first term and \(r\) is the common ratio.

In our case:
- First term \(a_1 = -1\)
- Common ratio \(r = -3\)

Thus, we can write the equation for the nth term \(a_n\) as:
\[ a_n = -1 \cdot (-3)^{n-1} \]

Finally, we can express the final equation as:
\[ a_n = -1 \cdot (-3)^{n-1} \]

If simplifying is needed to avoid a complex expression in a negative base, you would typically leave it as is. However, the required form can also be presented as:
\[ a_n = (-1)^{n} \cdot 3^{n-1} \]

Thus, following your request, the completed equation in the desired format is:
\[ a_n = -1 \cdot (-3)^{n-1} \]