Answer :
To find the formula that describes the given sequence, let's start by understanding the terms:
The sequence provided is:
- [tex]\(-2 \frac{2}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3}\)[/tex]
First, convert all mixed numbers to improper fractions to make the calculations easier:
1. [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
2. [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
3. [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
4. [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
5. [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
Next, let’s find the ratio between consecutive terms:
- The second term divided by the first term:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- The third term divided by the second term:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
- The fourth term divided by the third term:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2
\][/tex]
- The fifth term divided by the fourth term:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
We can observe that the ratio between each pair of consecutive terms is consistently [tex]\(2\)[/tex]. This suggests the sequence is a geometric sequence with a common ratio of [tex]\(2\)[/tex].
This means the formula that describes the sequence is:
[tex]\[ f(x+1) = 2 \cdot f(x) \][/tex]
Therefore, the correct answer is: [tex]\( f(x+1) = 2 \cdot f(x) \)[/tex]
The sequence provided is:
- [tex]\(-2 \frac{2}{3}\)[/tex]
- [tex]\(-5 \frac{1}{3}\)[/tex]
- [tex]\(-10 \frac{2}{3}\)[/tex]
- [tex]\(-21 \frac{1}{3}\)[/tex]
- [tex]\(-42 \frac{2}{3}\)[/tex]
First, convert all mixed numbers to improper fractions to make the calculations easier:
1. [tex]\(-2 \frac{2}{3} = -\frac{8}{3}\)[/tex]
2. [tex]\(-5 \frac{1}{3} = -\frac{16}{3}\)[/tex]
3. [tex]\(-10 \frac{2}{3} = -\frac{32}{3}\)[/tex]
4. [tex]\(-21 \frac{1}{3} = -\frac{64}{3}\)[/tex]
5. [tex]\(-42 \frac{2}{3} = -\frac{128}{3}\)[/tex]
Next, let’s find the ratio between consecutive terms:
- The second term divided by the first term:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
- The third term divided by the second term:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = 2
\][/tex]
- The fourth term divided by the third term:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = 2
\][/tex]
- The fifth term divided by the fourth term:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = 2
\][/tex]
We can observe that the ratio between each pair of consecutive terms is consistently [tex]\(2\)[/tex]. This suggests the sequence is a geometric sequence with a common ratio of [tex]\(2\)[/tex].
This means the formula that describes the sequence is:
[tex]\[ f(x+1) = 2 \cdot f(x) \][/tex]
Therefore, the correct answer is: [tex]\( f(x+1) = 2 \cdot f(x) \)[/tex]
