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Answer :
To solve this question, we need to determine which formula accurately describes the given sequence: [tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex].
### Step-by-Step Solution:
1. Convert Mixed Fractions to Improper Fractions:
- [tex]\(-2 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{8}{3}\)[/tex].
- [tex]\(-5 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{16}{3}\)[/tex].
- [tex]\(-10 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{32}{3}\)[/tex].
- [tex]\(-21 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{64}{3}\)[/tex].
- [tex]\(-42 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{128}{3}\)[/tex].
2. Identify the Pattern:
The transformed sequence now looks like: [tex]\(-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots\)[/tex].
3. Determine the Common Ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant, called the common ratio.
- Calculate the ratio between the second and first terms:
[tex]\[
\text{Ratio } = \frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- Similarly, calculate for the other pairs:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2, \quad \frac{-\frac{64}{3}}{-\frac{32}{3}} = 2, \quad \frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
All these calculations confirm that the common ratio is [tex]\(2\)[/tex].
4. Select the Correct Formula:
The formula that represents this pattern in the sequence is:
- [tex]\(f(x+1) = 2f(x)\)[/tex]
Therefore, the sequence is described by the formula [tex]\(f(x+1) = 2f(x)\)[/tex].
### Step-by-Step Solution:
1. Convert Mixed Fractions to Improper Fractions:
- [tex]\(-2 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{8}{3}\)[/tex].
- [tex]\(-5 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{16}{3}\)[/tex].
- [tex]\(-10 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{32}{3}\)[/tex].
- [tex]\(-21 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{64}{3}\)[/tex].
- [tex]\(-42 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{128}{3}\)[/tex].
2. Identify the Pattern:
The transformed sequence now looks like: [tex]\(-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots\)[/tex].
3. Determine the Common Ratio:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant, called the common ratio.
- Calculate the ratio between the second and first terms:
[tex]\[
\text{Ratio } = \frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- Similarly, calculate for the other pairs:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2, \quad \frac{-\frac{64}{3}}{-\frac{32}{3}} = 2, \quad \frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
All these calculations confirm that the common ratio is [tex]\(2\)[/tex].
4. Select the Correct Formula:
The formula that represents this pattern in the sequence is:
- [tex]\(f(x+1) = 2f(x)\)[/tex]
Therefore, the sequence is described by the formula [tex]\(f(x+1) = 2f(x)\)[/tex].
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