Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


2 , –8 , 32 , ...


Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To find the equation for the given sequence \(2, -8, 32, \ldots\), we can first analyze the sequence to determine the pattern.

Let's examine the terms:

- \(a_1 = 2\)
- \(a_2 = -8\)
- \(a_3 = 32\)

Notice that the terms seem to alternate in sign and increase in absolute value.

Next, we can analyze the absolute values of the terms:

- \(|a_1| = 2\)
- \(|a_2| = 8\)
- \(|a_3| = 32\)

It looks like the absolute values can be represented as powers of \(2\):

- \(2 = 2^1\)
- \(8 = 2^3\) (which is \(2^3\))
- \(32 = 2^5\) (which is \(2^5\))

The exponents of 2 seem to follow the pattern \(2n - 1\) for \(n = 1, 2, 3, \ldots\):

- For \(n = 1\), \(2(1) - 1 = 1\)
- For \(n = 2\), \(2(2) - 1 = 3\)
- For \(n = 3\), \(2(3) - 1 = 5\)

Thus, the absolute value of the \(n\)-th term can be represented as:

\[
|a_n| = 2^{2n - 1}
\]

Next, we introduce the alternating sign:

The terms \(a_n\) are positive for odd \(n\) and negative for even \(n\). We can use \((-1)^{n+1}\) to alternate the sign:

\[
a_n = (-1)^{n+1} \cdot 2^{2n - 1}
\]

Putting it together, the final equation is:

\[
\boxed{a_n = (-1)^{n+1} \cdot 2^{2n - 1}}
\]