To determine which formula can be used to describe the given sequence, let's analyze the sequence step-by-step.
Consider the sequence in the mixed fraction form:
[tex]\[ -2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots \][/tex]
First, let's convert each term from mixed fractions to improper fractions:
1. [tex]\( -2 \frac{2}{3} = -\frac{6+2}{3} = -\frac{8}{3} \)[/tex]
2. [tex]\( -5 \frac{1}{3} = -\frac{15+1}{3} = -\frac{16}{3} \)[/tex]
3. [tex]\( -10 \frac{2}{3} = -\frac{30+2}{3} = -\frac{32}{3} \)[/tex]
4. [tex]\( -21 \frac{1}{3} = -\frac{63+1}{3} = -\frac{64}{3} \)[/tex]
5. [tex]\( -42 \frac{2}{3} = -\frac{126+2}{3} = -\frac{128}{3} \)[/tex]
So the sequence now looks like:
[tex]\[ -\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots \][/tex]
Observing the fractions, we see that with each term, the numerator is doubling:
[tex]\[ -8, -16, -32, -64, -128, \ldots \][/tex]
This indicates that each term is obtained by multiplying the previous term by 2. Now, considering the improper fractions, we can generalize this pattern into a function, [tex]\( f \)[/tex], where:
[tex]\[ f(x+1) = 2 f(x) \][/tex]
Let's verify this:
- [tex]\( f(1) = -\frac{8}{3} \)[/tex]
- [tex]\( f(2) = 2 \cdot f(1) = 2 \cdot -\frac{8}{3} = -\frac{16}{3} \)[/tex]
- [tex]\( f(3) = 2 \cdot f(2) = 2 \cdot -\frac{16}{3} = -\frac{32}{3} \)[/tex]
- [tex]\( f(4) = 2 \cdot f(3) = 2 \cdot -\frac{32}{3} = -\frac{64}{3} \)[/tex]
- [tex]\( f(5) = 2 \cdot f(4) = 2 \cdot -\frac{64}{3} = -\frac{128}{3} \)[/tex]
This matches our sequence perfectly, confirming that each term is obtained by multiplying the previous term by 2.
Therefore, the correct formula to describe the sequence is:
[tex]\[ f(x+1) = 2 f(x) \][/tex]
The correct answer is:
[tex]\[ \boxed{f(x+1) = 2 f(x)} \][/tex]
