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Answer :
To solve the problem of finding the formula that describes the given sequence, let's first look at the sequence provided:
[tex]\[
-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \cdots
\][/tex]
This sequence seems to be decreasing with each term, and they can be expressed as improper fractions for easier comparison:
1. Convert the mixed numbers to improper fractions:
- [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]
2. Observe how the sequence progresses:
- To find if there is a consistent multiplication factor between terms, divide each term by its predecessor:
- [tex]\(\frac{-16/3}{-8/3} = 2\)[/tex]
- [tex]\(\frac{-32/3}{-16/3} = 2\)[/tex]
- [tex]\(\frac{-64/3}{-32/3} = 2\)[/tex]
- [tex]\(\frac{-128/3}{-64/3} = 2\)[/tex]
3. Each term is obtained by multiplying the previous term by [tex]\(-2\)[/tex]. Therefore, there is a consistent pattern that each term becomes twice the previous term but in the negative direction. The rule for this sequence is:
[tex]\[
f(x+1) = -2 \cdot f(x)
\][/tex]
Thus, the formula that accurately describes this sequence is [tex]\(f(x+1) = -2 \cdot f(x)\)[/tex].
[tex]\[
-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \cdots
\][/tex]
This sequence seems to be decreasing with each term, and they can be expressed as improper fractions for easier comparison:
1. Convert the mixed numbers to improper fractions:
- [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]
2. Observe how the sequence progresses:
- To find if there is a consistent multiplication factor between terms, divide each term by its predecessor:
- [tex]\(\frac{-16/3}{-8/3} = 2\)[/tex]
- [tex]\(\frac{-32/3}{-16/3} = 2\)[/tex]
- [tex]\(\frac{-64/3}{-32/3} = 2\)[/tex]
- [tex]\(\frac{-128/3}{-64/3} = 2\)[/tex]
3. Each term is obtained by multiplying the previous term by [tex]\(-2\)[/tex]. Therefore, there is a consistent pattern that each term becomes twice the previous term but in the negative direction. The rule for this sequence is:
[tex]\[
f(x+1) = -2 \cdot f(x)
\][/tex]
Thus, the formula that accurately describes this sequence is [tex]\(f(x+1) = -2 \cdot f(x)\)[/tex].
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