Answer :
To determine which formula fits the given sequence, let's analyze the sequence step by step:
The sequence provided is:
[tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex]
1. Convert 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]
Now, the sequence is: [tex]\(-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots\)[/tex]
2. Find the Common Ratio Between Consecutive Terms:
To determine the relationship between the terms, divide each term by the previous term:
- Ratio between 2nd and 1st terms: [tex]\(\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2.0\)[/tex]
- Ratio between 3rd and 2nd terms: [tex]\(\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2.0\)[/tex]
- Ratio between 4th and 3rd terms: [tex]\(\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2.0\)[/tex]
- Ratio between 5th and 4th terms: [tex]\(\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2.0\)[/tex]
All the ratios are consistent and equal to 2.
3. Determine the Formula Based on the Consistent Ratio:
Since the ratio is consistently 2, this means each term is twice the previous term. This corresponds to the formula:
[tex]\[
f(x+1) = 2 \times f(x)
\][/tex]
Therefore, the formula that can be used to describe the sequence is [tex]\(f(x+1) = 2 f(x)\)[/tex].
The sequence provided is:
[tex]\(-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots\)[/tex]
1. Convert 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]
Now, the sequence is: [tex]\(-\frac{8}{3}, -\frac{16}{3}, -\frac{32}{3}, -\frac{64}{3}, -\frac{128}{3}, \ldots\)[/tex]
2. Find the Common Ratio Between Consecutive Terms:
To determine the relationship between the terms, divide each term by the previous term:
- Ratio between 2nd and 1st terms: [tex]\(\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2.0\)[/tex]
- Ratio between 3rd and 2nd terms: [tex]\(\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2.0\)[/tex]
- Ratio between 4th and 3rd terms: [tex]\(\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2.0\)[/tex]
- Ratio between 5th and 4th terms: [tex]\(\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2.0\)[/tex]
All the ratios are consistent and equal to 2.
3. Determine the Formula Based on the Consistent Ratio:
Since the ratio is consistently 2, this means each term is twice the previous term. This corresponds to the formula:
[tex]\[
f(x+1) = 2 \times f(x)
\][/tex]
Therefore, the formula that can be used to describe the sequence is [tex]\(f(x+1) = 2 f(x)\)[/tex].
