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, -2, -4, ...
Write your answer using decimals and integers.
an = ____ (___)^n-1

Answer 1

To find an equation for the sequence \(-1, -2, -4, ...\), we first need to observe the pattern in the terms provided.

1. The first term (\(n=1\)) is \(-1\).
2. The second term (\(n=2\)) is \(-2\).
3. The third term (\(n=3\)) is \(-4\).

If we look at the absolute values of the terms, we have:
- First term: 1
- Second term: 2
- Third term: 4

The absolute values double with each term, so this suggests a geometric progression with a common ratio of 2.

Thus, we can express the absolute value of the \(n\)-th term as:
\[
|a_n| = 2^{n-1}
\]

Since each term in the original sequence is negative, we can express \(a_n\) as:
\[
a_n = -2^{n-1}
\]

Therefore, the equation that describes the sequence is:
\[
a_n = - (2^{n-1})
\]

In the format you've requested:
\[
a_n = -1 \cdot (2)^{n-1}
\]