Consider the sequence: [tex]-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots[/tex]

Which formula can be used to describe the sequence?

A. [tex]f(x+1) = -2 f(x)[/tex]

B. [tex]f(x+1) = -\frac{1}{2} f(x)[/tex]

C. [tex]f(x+1) = \frac{1}{2} f(x)[/tex]

D. [tex]f(x+1) = 2 f(x)[/tex]

To find the formula that describes the given sequence, let's analyze the terms provided:

1. The sequence provided is:
[tex]$-2 \frac{2}{3},\ -5 \frac{1}{3},\ -10 \frac{2}{3},\ -21 \frac{1}{3},\ -42 \frac{2}{3}, \ldots$[/tex]

2. First, let's convert each mixed number to an improper fraction:

- [tex]$-2 \frac{2}{3} = \frac{-8}{3}$[/tex]
- [tex]$-5 \frac{1}{3} = \frac{-16}{3}$[/tex]
- [tex]$-10 \frac{2}{3} = \frac{-32}{3}$[/tex]
- [tex]$-21 \frac{1}{3} = \frac{-64}{3}$[/tex]
- [tex]$-42 \frac{2}{3} = \frac{-128}{3}$[/tex]

3. Now, let's look for a pattern. Notice how each term appears to be multiplied by a constant to get the next term:

[tex]\[
\frac{-8}{3} \times (-2) = \frac{-16}{3}
\][/tex]
[tex]\[
\frac{-16}{3} \times (-2) = \frac{-32}{3}
\][/tex]
[tex]\[
\frac{-32}{3} \times (-2) = \frac{-64}{3}
\][/tex]
[tex]\[
\frac{-64}{3} \times (-2) = \frac{-128}{3}
\][/tex]

Each term is obtained by multiplying the previous term by [tex]$-2$[/tex].

4. Therefore, the formula that can be used to describe the sequence is:
[tex]$f(x+1) = -2 f(x)$[/tex]

This formula matches the pattern of the sequence, where each term is [tex]$-2$[/tex] times the previous term.