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Answer :
We start with the sequence:
[tex]$$-5 \tfrac{1}{3},\quad -10 \tfrac{2}{3},\quad -21 \tfrac{1}{3},\quad -42 \tfrac{2}{3},\quad \ldots$$[/tex]
Step 1. Convert mixed numbers to improper fractions.
For the first term:
[tex]$$-5 \tfrac{1}{3} = -\frac{(5 \times 3 + 1)}{3} = -\frac{16}{3}.$$[/tex]
For the second term:
[tex]$$-10 \tfrac{2}{3} = -\frac{(10 \times 3 + 2)}{3} = -\frac{32}{3}.$$[/tex]
For the third term:
[tex]$$-21 \tfrac{1}{3} = -\frac{(21 \times 3 + 1)}{3} = -\frac{64}{3}.$$[/tex]
For the fourth term:
[tex]$$-42 \tfrac{2}{3} = -\frac{(42 \times 3 + 2)}{3} = -\frac{128}{3}.$$[/tex]
Thus, the sequence in improper fraction form is:
[tex]$$-\frac{16}{3},\quad -\frac{32}{3},\quad -\frac{64}{3},\quad -\frac{128}{3},\quad \ldots$$[/tex]
Step 2. Identify the pattern.
Notice that each term is obtained by multiplying the previous term by 2:
[tex]$$-\frac{16}{3} \times 2 = -\frac{32}{3},$$[/tex]
[tex]$$-\frac{32}{3} \times 2 = -\frac{64}{3},$$[/tex]
[tex]$$-\frac{64}{3} \times 2 = -\frac{128}{3}.$$[/tex]
Thus, the relationship between consecutive terms is given by:
[tex]$$f(n+1) = 2 \, f(n).$$[/tex]
Step 3. Conclude the appropriate formula.
From the options provided, the recurrence relation corresponding to the pattern is:
[tex]$$f(x+1)=2 f(x).$$[/tex]
Therefore, the correct formula is:
[tex]$$\boxed{f(x+1)=2 f(x)}.$$[/tex]
[tex]$$-5 \tfrac{1}{3},\quad -10 \tfrac{2}{3},\quad -21 \tfrac{1}{3},\quad -42 \tfrac{2}{3},\quad \ldots$$[/tex]
Step 1. Convert mixed numbers to improper fractions.
For the first term:
[tex]$$-5 \tfrac{1}{3} = -\frac{(5 \times 3 + 1)}{3} = -\frac{16}{3}.$$[/tex]
For the second term:
[tex]$$-10 \tfrac{2}{3} = -\frac{(10 \times 3 + 2)}{3} = -\frac{32}{3}.$$[/tex]
For the third term:
[tex]$$-21 \tfrac{1}{3} = -\frac{(21 \times 3 + 1)}{3} = -\frac{64}{3}.$$[/tex]
For the fourth term:
[tex]$$-42 \tfrac{2}{3} = -\frac{(42 \times 3 + 2)}{3} = -\frac{128}{3}.$$[/tex]
Thus, the sequence in improper fraction form is:
[tex]$$-\frac{16}{3},\quad -\frac{32}{3},\quad -\frac{64}{3},\quad -\frac{128}{3},\quad \ldots$$[/tex]
Step 2. Identify the pattern.
Notice that each term is obtained by multiplying the previous term by 2:
[tex]$$-\frac{16}{3} \times 2 = -\frac{32}{3},$$[/tex]
[tex]$$-\frac{32}{3} \times 2 = -\frac{64}{3},$$[/tex]
[tex]$$-\frac{64}{3} \times 2 = -\frac{128}{3}.$$[/tex]
Thus, the relationship between consecutive terms is given by:
[tex]$$f(n+1) = 2 \, f(n).$$[/tex]
Step 3. Conclude the appropriate formula.
From the options provided, the recurrence relation corresponding to the pattern is:
[tex]$$f(x+1)=2 f(x).$$[/tex]
Therefore, the correct formula is:
[tex]$$\boxed{f(x+1)=2 f(x)}.$$[/tex]
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