Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.
–69 , 138 , –276 , ...
Write your answer using decimals and integers.

an = _(_)^ n^ -^ 1

Answer 1

To find an equation to describe the given sequence (-69, 138, -276, ...), we first need to identify the pattern or rule governing the sequence.

Let's analyze the terms:

1. First term (n = 1): a1 = -69
2. Second term (n = 2): a2 = 138
3. Third term (n = 3): a3 = -276

Now let's look at the ratio of consecutive terms:

- From the 1st to the 2nd term:
a2 / a1 = 138 / -69 = -2
- From the 2nd to the 3rd term:
a3 / a2 = -276 / 138 = -2

This consistent ratio suggests that the sequence is geometric with a common ratio of -2.

The general formula for a geometric sequence can be expressed as:

\[ a_n = a_1 \cdot r^{(n-1)} \]

where:
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the term number.

From our observations:
- \( a_1 = -69 \)
- \( r = -2 \)

Plugging these values into the formula gives:

\[ a_n = -69 \cdot (-2)^{(n-1)} \]

So, the final equation to describe the sequence is:

\[ a_n = -69 \cdot (-2)^{(n-1)} \]

Thus, in the format you requested:

**an = -69(-2)^(n - 1)**