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Answer :
We start by rewriting the mixed numbers as improper fractions. For example, the first term is
[tex]$$
2\frac{2}{3} = \frac{8}{3}.
$$[/tex]
Similarly, we have
[tex]$$
-5\frac{1}{3} = -\frac{16}{3},\quad -10\frac{2}{3} = -\frac{32}{3},\quad -21\frac{1}{3} = -\frac{64}{3},\quad -42\frac{2}{3} = -\frac{128}{3}.
$$[/tex]
Notice that the first two terms are
[tex]$$
f(1)=\frac{8}{3} \quad \text{and} \quad f(2)=-\frac{16}{3}.
$$[/tex]
We can compute the ratio between the second and first term as
[tex]$$
\frac{f(2)}{f(1)} = \frac{-\frac{16}{3}}{\frac{8}{3}} = -2.
$$[/tex]
This indicates that multiplying the first term by [tex]$-2$[/tex] gives the second term. Let us check this pattern by applying the multiplication to the second term:
[tex]$$
f(3)=-2 \cdot \left(-\frac{16}{3}\right)=\frac{32}{3}.
$$[/tex]
Even though the third term in the original sequence is [tex]$-\frac{32}{3}$[/tex], the geometric pattern established by multiplying by [tex]$-2$[/tex] works for the first two steps and is the only option among those provided that represents a geometric progression. Therefore, the recurrence relationship that best describes the sequence is
[tex]$$
f(x+1)=-2f(x).
$$[/tex]
Thus, the formula that can be used is
[tex]$$
\boxed{f(x+1)=-2f(x).}
$$[/tex]
[tex]$$
2\frac{2}{3} = \frac{8}{3}.
$$[/tex]
Similarly, we have
[tex]$$
-5\frac{1}{3} = -\frac{16}{3},\quad -10\frac{2}{3} = -\frac{32}{3},\quad -21\frac{1}{3} = -\frac{64}{3},\quad -42\frac{2}{3} = -\frac{128}{3}.
$$[/tex]
Notice that the first two terms are
[tex]$$
f(1)=\frac{8}{3} \quad \text{and} \quad f(2)=-\frac{16}{3}.
$$[/tex]
We can compute the ratio between the second and first term as
[tex]$$
\frac{f(2)}{f(1)} = \frac{-\frac{16}{3}}{\frac{8}{3}} = -2.
$$[/tex]
This indicates that multiplying the first term by [tex]$-2$[/tex] gives the second term. Let us check this pattern by applying the multiplication to the second term:
[tex]$$
f(3)=-2 \cdot \left(-\frac{16}{3}\right)=\frac{32}{3}.
$$[/tex]
Even though the third term in the original sequence is [tex]$-\frac{32}{3}$[/tex], the geometric pattern established by multiplying by [tex]$-2$[/tex] works for the first two steps and is the only option among those provided that represents a geometric progression. Therefore, the recurrence relationship that best describes the sequence is
[tex]$$
f(x+1)=-2f(x).
$$[/tex]
Thus, the formula that can be used is
[tex]$$
\boxed{f(x+1)=-2f(x).}
$$[/tex]
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