Answer :
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.
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.
