Answer :
The sequence given is
[tex]$$-2 \frac{2}{3},\ -5 \frac{1}{3},\ -10 \frac{2}{3},\ -21 \frac{1}{3},\ -42 \frac{2}{3},\ \dots$$[/tex]
Step 1. Convert the mixed numbers to improper fractions.
For each term:
- [tex]$$-2 \frac{2}{3} = -\frac{2 \times 3 + 2}{3} = -\frac{8}{3}$$[/tex]
- [tex]$$-5 \frac{1}{3} = -\frac{5 \times 3 + 1}{3} = -\frac{16}{3}$$[/tex]
- [tex]$$-10 \frac{2}{3} = -\frac{10 \times 3 + 2}{3} = -\frac{32}{3}$$[/tex]
- [tex]$$-21 \frac{1}{3} = -\frac{21 \times 3 + 1}{3} = -\frac{64}{3}$$[/tex]
- [tex]$$-42 \frac{2}{3} = -\frac{42 \times 3 + 2}{3} = -\frac{128}{3}$$[/tex]
Step 2. Determine the common ratio.
We find the ratio between successive terms. For instance, the ratio between the second term and the first term is
[tex]$$\frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16/3}{8/3} = 2.$$[/tex]
Likewise, the ratio between the third and the second term is
[tex]$$\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2.$$[/tex]
All successive ratios are equal to 2.
Step 3. Write the formula for the sequence.
When each term is obtained by multiplying the previous term by 2, the sequence can be defined by the recursive formula
[tex]$$f(x+1) = 2 \, f(x).$$[/tex]
Thus, the formula that describes the sequence is
[tex]$$\boxed{f(x+1) = 2 \, f(x)}.$$[/tex]
[tex]$$-2 \frac{2}{3},\ -5 \frac{1}{3},\ -10 \frac{2}{3},\ -21 \frac{1}{3},\ -42 \frac{2}{3},\ \dots$$[/tex]
Step 1. Convert the mixed numbers to improper fractions.
For each term:
- [tex]$$-2 \frac{2}{3} = -\frac{2 \times 3 + 2}{3} = -\frac{8}{3}$$[/tex]
- [tex]$$-5 \frac{1}{3} = -\frac{5 \times 3 + 1}{3} = -\frac{16}{3}$$[/tex]
- [tex]$$-10 \frac{2}{3} = -\frac{10 \times 3 + 2}{3} = -\frac{32}{3}$$[/tex]
- [tex]$$-21 \frac{1}{3} = -\frac{21 \times 3 + 1}{3} = -\frac{64}{3}$$[/tex]
- [tex]$$-42 \frac{2}{3} = -\frac{42 \times 3 + 2}{3} = -\frac{128}{3}$$[/tex]
Step 2. Determine the common ratio.
We find the ratio between successive terms. For instance, the ratio between the second term and the first term is
[tex]$$\frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16/3}{8/3} = 2.$$[/tex]
Likewise, the ratio between the third and the second term is
[tex]$$\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2.$$[/tex]
All successive ratios are equal to 2.
Step 3. Write the formula for the sequence.
When each term is obtained by multiplying the previous term by 2, the sequence can be defined by the recursive formula
[tex]$$f(x+1) = 2 \, f(x).$$[/tex]
Thus, the formula that describes the sequence is
[tex]$$\boxed{f(x+1) = 2 \, f(x)}.$$[/tex]
