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Answer :
To determine which formula can describe 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]
we'll analyze the pattern between the numbers.
1. Convert Mixed Numbers to Improper Fractions
- The sequence is:
[tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}\)[/tex].
- Convert to improper fractions:
- [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
2. Find the Ratio Between Consecutive Terms
- Calculate the ratio between each pair of consecutive terms:
- Ratio from [tex]\(-\frac{8}{3}\)[/tex] to [tex]\(-\frac{16}{3}\)[/tex]:
[tex]\(\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2\)[/tex]
- Ratio from [tex]\(-\frac{16}{3}\)[/tex] to [tex]\(-\frac{32}{3}\)[/tex]:
[tex]\(\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2\)[/tex]
- Ratio from [tex]\(-\frac{32}{3}\)[/tex] to [tex]\(-\frac{64}{3}\)[/tex]:
[tex]\(\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2\)[/tex]
- Ratio from [tex]\(-\frac{64}{3}\)[/tex] to [tex]\(-\frac{128}{3}\)[/tex]:
[tex]\(\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2\)[/tex]
3. Identify the Common Ratio
- All the ratios between consecutive terms are the same: [tex]\(2\)[/tex].
4. Determine the Formula
- Since the ratio is consistent and equal to 2, we use this ratio to determine the formula for [tex]\(f(x+1)\)[/tex].
- The formula for the sequence is:
[tex]\(f(x+1) = 2f(x)\)[/tex].
Therefore, the correct formula that describes the sequence is [tex]\(f(x+1) = 2f(x)\)[/tex].
[tex]$$-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots$$[/tex]
we'll analyze the pattern between the numbers.
1. Convert Mixed Numbers to Improper Fractions
- The sequence is:
[tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}\)[/tex].
- Convert to improper fractions:
- [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
2. Find the Ratio Between Consecutive Terms
- Calculate the ratio between each pair of consecutive terms:
- Ratio from [tex]\(-\frac{8}{3}\)[/tex] to [tex]\(-\frac{16}{3}\)[/tex]:
[tex]\(\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2\)[/tex]
- Ratio from [tex]\(-\frac{16}{3}\)[/tex] to [tex]\(-\frac{32}{3}\)[/tex]:
[tex]\(\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2\)[/tex]
- Ratio from [tex]\(-\frac{32}{3}\)[/tex] to [tex]\(-\frac{64}{3}\)[/tex]:
[tex]\(\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2\)[/tex]
- Ratio from [tex]\(-\frac{64}{3}\)[/tex] to [tex]\(-\frac{128}{3}\)[/tex]:
[tex]\(\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2\)[/tex]
3. Identify the Common Ratio
- All the ratios between consecutive terms are the same: [tex]\(2\)[/tex].
4. Determine the Formula
- Since the ratio is consistent and equal to 2, we use this ratio to determine the formula for [tex]\(f(x+1)\)[/tex].
- The formula for the sequence is:
[tex]\(f(x+1) = 2f(x)\)[/tex].
Therefore, the correct formula that describes the sequence is [tex]\(f(x+1) = 2f(x)\)[/tex].
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