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Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

–6 , –12 , –24 , ...

Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To find an equation that describes the sequence \(-6, -12, -24, \ldots\), we first need to determine a pattern.

Let's look for a relationship between the terms and their positions \(n\):

1. The first term (\(n=1\)):

   • \(a_1 = -6\)

2. The second term (\(n=2\)):

   • \(a_2 = -12\)

3. The third term (\(n=3\)):

   • \(a_3 = -24\)

Now, we can observe how each term is related to the previous one:

• From \(-6\) to \(-12\), it is multiplied by \(2\) (i.e. \(-6 \times 2 = -12\)).
• From \(-12\) to \(-24\), it is again multiplied by \(2\) (i.e. \(-12 \times 2 = -24\)).

This indicates that the terms are multiplying by 2 as we move to the next term.

Now let’s denote the formula for the \(n\)-th term \(a_n\). The first term can be represented as:

• \(a_1 = -6 = -6 \times 2^{0}\) The second term is:
• \(a_2 = -12 = -6 \times 2^{1}\) The third term is:
• \(a_3 = -24 = -6 \times 2^{2}\)

From this, we can see that: \[ a_n = -6 \times 2^{n-1} \]

Thus, the expression that describes the sequence is: \[ a_n = -6 \times 2^{n-1} \]