Answer :
We start with the sequence
[tex]$$
-2\frac{2}{3},\quad -5\frac{1}{3},\quad -10\frac{2}{3},\quad -21\frac{1}{3},\quad -42\frac{2}{3},\quad \ldots
$$[/tex]
Step 1. Convert mixed numbers to improper fractions.
For each term, we write it as an improper fraction:
- The first term:
[tex]$$
-2\frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\frac{6+2}{3} = -\frac{8}{3}.
$$[/tex]
- The second term:
[tex]$$
-5\frac{1}{3} = -\left(5 + \frac{1}{3}\right) = -\frac{15+1}{3} = -\frac{16}{3}.
$$[/tex]
- The third term:
[tex]$$
-10\frac{2}{3} = -\left(10 + \frac{2}{3}\right) = -\frac{30+2}{3} = -\frac{32}{3}.
$$[/tex]
- The fourth term:
[tex]$$
-21\frac{1}{3} = -\left(21 + \frac{1}{3}\right) = -\frac{63+1}{3} = -\frac{64}{3}.
$$[/tex]
- The fifth term:
[tex]$$
-42\frac{2}{3} = -\left(42 + \frac{2}{3}\right) = -\frac{126+2}{3} = -\frac{128}{3}.
$$[/tex]
So, the sequence in improper fractions is
[tex]$$
-\frac{8}{3}, \quad -\frac{16}{3}, \quad -\frac{32}{3}, \quad -\frac{64}{3}, \quad -\frac{128}{3}, \quad \ldots
$$[/tex]
Step 2. Determine the pattern between consecutive terms.
We look at the ratio of a term to its preceding term. For example, compute the ratio of the second term to the first term:
[tex]$$
\text{Ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2.
$$[/tex]
Check another pair:
[tex]$$
\text{Ratio} = \frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2.
$$[/tex]
Since the ratio between consecutive terms is always [tex]$2$[/tex], the sequence is geometric with common ratio [tex]$2$[/tex].
Step 3. Write the recurrence relation.
For a geometric sequence, each term is obtained by multiplying the previous term by the common ratio. The general recurrence relation is
[tex]$$
f(x+1) = r \cdot f(x).
$$[/tex]
Since [tex]$r = 2$[/tex], we have
[tex]$$
f(x+1) = 2f(x).
$$[/tex]
Conclusion:
The formula that describes the sequence is
[tex]$$
f(x+1) = 2f(x).
$$[/tex]
[tex]$$
-2\frac{2}{3},\quad -5\frac{1}{3},\quad -10\frac{2}{3},\quad -21\frac{1}{3},\quad -42\frac{2}{3},\quad \ldots
$$[/tex]
Step 1. Convert mixed numbers to improper fractions.
For each term, we write it as an improper fraction:
- The first term:
[tex]$$
-2\frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\frac{6+2}{3} = -\frac{8}{3}.
$$[/tex]
- The second term:
[tex]$$
-5\frac{1}{3} = -\left(5 + \frac{1}{3}\right) = -\frac{15+1}{3} = -\frac{16}{3}.
$$[/tex]
- The third term:
[tex]$$
-10\frac{2}{3} = -\left(10 + \frac{2}{3}\right) = -\frac{30+2}{3} = -\frac{32}{3}.
$$[/tex]
- The fourth term:
[tex]$$
-21\frac{1}{3} = -\left(21 + \frac{1}{3}\right) = -\frac{63+1}{3} = -\frac{64}{3}.
$$[/tex]
- The fifth term:
[tex]$$
-42\frac{2}{3} = -\left(42 + \frac{2}{3}\right) = -\frac{126+2}{3} = -\frac{128}{3}.
$$[/tex]
So, the sequence in improper fractions is
[tex]$$
-\frac{8}{3}, \quad -\frac{16}{3}, \quad -\frac{32}{3}, \quad -\frac{64}{3}, \quad -\frac{128}{3}, \quad \ldots
$$[/tex]
Step 2. Determine the pattern between consecutive terms.
We look at the ratio of a term to its preceding term. For example, compute the ratio of the second term to the first term:
[tex]$$
\text{Ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2.
$$[/tex]
Check another pair:
[tex]$$
\text{Ratio} = \frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2.
$$[/tex]
Since the ratio between consecutive terms is always [tex]$2$[/tex], the sequence is geometric with common ratio [tex]$2$[/tex].
Step 3. Write the recurrence relation.
For a geometric sequence, each term is obtained by multiplying the previous term by the common ratio. The general recurrence relation is
[tex]$$
f(x+1) = r \cdot f(x).
$$[/tex]
Since [tex]$r = 2$[/tex], we have
[tex]$$
f(x+1) = 2f(x).
$$[/tex]
Conclusion:
The formula that describes the sequence is
[tex]$$
f(x+1) = 2f(x).
$$[/tex]
