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Answer :
To find the formula describing the sequence given by the terms [tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex], we can follow these steps:
1. Convert the Mixed Numbers 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. Identify the Pattern:
- Look at the sequence of improper fractions: [tex]\(-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}\)[/tex].
3. Find the Common Ratio:
- To see if this is a geometric sequence, divide each term by the previous one.
- [tex]\( \text{Common ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{3} \div \frac{8}{3} = 2 \)[/tex]
- Confirm the common ratio by checking:
- [tex]\( \frac{-\frac{32}{3}}{-\frac{16}{3}} = 2 \)[/tex]
- [tex]\( \frac{-\frac{64}{3}}{-\frac{32}{3}} = 2 \)[/tex]
- [tex]\( \frac{-\frac{128}{3}}{-\frac{64}{3}} = 2 \)[/tex]
4. Conclusion:
- The sequence follows a consistent pattern where each term is multiplied by 2 to get the next term.
- Therefore, the correct formula to describe this sequence is [tex]\( f(x+1) = 2f(x) \)[/tex].
Thus, the formula that can be used to describe the sequence is [tex]\( f(x+1) = 2 f(x) \)[/tex].
1. Convert the Mixed Numbers 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. Identify the Pattern:
- Look at the sequence of improper fractions: [tex]\(-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}\)[/tex].
3. Find the Common Ratio:
- To see if this is a geometric sequence, divide each term by the previous one.
- [tex]\( \text{Common ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{3} \div \frac{8}{3} = 2 \)[/tex]
- Confirm the common ratio by checking:
- [tex]\( \frac{-\frac{32}{3}}{-\frac{16}{3}} = 2 \)[/tex]
- [tex]\( \frac{-\frac{64}{3}}{-\frac{32}{3}} = 2 \)[/tex]
- [tex]\( \frac{-\frac{128}{3}}{-\frac{64}{3}} = 2 \)[/tex]
4. Conclusion:
- The sequence follows a consistent pattern where each term is multiplied by 2 to get the next term.
- Therefore, the correct formula to describe this sequence is [tex]\( f(x+1) = 2f(x) \)[/tex].
Thus, the formula that can be used to describe the sequence is [tex]\( f(x+1) = 2 f(x) \)[/tex].
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