Answer :
First, convert the mixed numbers into improper fractions:
- The first term is
[tex]$$-2\frac{2}{3} = -\frac{8}{3}.$$[/tex]
- The second term is
[tex]$$-5\frac{1}{3} = -\frac{16}{3}.$$[/tex]
- The third term is
[tex]$$-10\frac{2}{3} = -\frac{32}{3}.$$[/tex]
- The fourth term is
[tex]$$-21\frac{1}{3} = -\frac{64}{3}.$$[/tex]
- The fifth term is
[tex]$$-42\frac{2}{3} = -\frac{128}{3}.$$[/tex]
Now, find the common ratio by dividing the second term by the first term:
[tex]$$
\text{ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2.
$$[/tex]
Since each term is obtained by multiplying the previous term by [tex]$2$[/tex], the sequence is geometric with a common ratio of [tex]$2$[/tex]. Therefore, the recurrence relation for the sequence is
[tex]$$
f(x+1) = 2 \, f(x).
$$[/tex]
Comparing with the given answer choices, the correct formula is
[tex]$$
f(x+1)=2 f(x).
$$[/tex]
Thus, the answer is choice 4.
- The first term is
[tex]$$-2\frac{2}{3} = -\frac{8}{3}.$$[/tex]
- The second term is
[tex]$$-5\frac{1}{3} = -\frac{16}{3}.$$[/tex]
- The third term is
[tex]$$-10\frac{2}{3} = -\frac{32}{3}.$$[/tex]
- The fourth term is
[tex]$$-21\frac{1}{3} = -\frac{64}{3}.$$[/tex]
- The fifth term is
[tex]$$-42\frac{2}{3} = -\frac{128}{3}.$$[/tex]
Now, find the common ratio by dividing the second term by the first term:
[tex]$$
\text{ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2.
$$[/tex]
Since each term is obtained by multiplying the previous term by [tex]$2$[/tex], the sequence is geometric with a common ratio of [tex]$2$[/tex]. Therefore, the recurrence relation for the sequence is
[tex]$$
f(x+1) = 2 \, f(x).
$$[/tex]
Comparing with the given answer choices, the correct formula is
[tex]$$
f(x+1)=2 f(x).
$$[/tex]
Thus, the answer is choice 4.
