Answer :
To determine the formula that describes the 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 need to analyze the pattern or rule that connects the sequence's terms.
First, let's express these numbers as improper fractions to make the pattern easier to see:
1. [tex]\(-2 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{8}{3}\)[/tex].
2. [tex]\(-5 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{16}{3}\)[/tex].
3. [tex]\(-10 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{32}{3}\)[/tex].
4. [tex]\(-21 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{64}{3}\)[/tex].
5. [tex]\(-42 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{128}{3}\)[/tex].
Next, let's examine the ratio between consecutive terms to see if there is a geometric pattern:
- The ratio of the second term to the first term is:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2
\][/tex]
- The ratio of the third term to the second term is:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2
\][/tex]
- The ratio of the fourth term to the third term is:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = \frac{64}{32} = 2
\][/tex]
- The ratio of the fifth term to the fourth term is:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = \frac{128}{64} = 2
\][/tex]
In each case, the ratio is the same, which is 2. This tells us that the sequence is geometric with a common ratio of 2.
Therefore, the formula to describe this sequence is:
[tex]\[ f(x+1) = 2 f(x) \][/tex]
This means that each term in the sequence is twice the previous term.
First, let's express these numbers as improper fractions to make the pattern easier to see:
1. [tex]\(-2 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{8}{3}\)[/tex].
2. [tex]\(-5 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{16}{3}\)[/tex].
3. [tex]\(-10 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{32}{3}\)[/tex].
4. [tex]\(-21 \frac{1}{3}\)[/tex] becomes [tex]\(-\frac{64}{3}\)[/tex].
5. [tex]\(-42 \frac{2}{3}\)[/tex] becomes [tex]\(-\frac{128}{3}\)[/tex].
Next, let's examine the ratio between consecutive terms to see if there is a geometric pattern:
- The ratio of the second term to the first term is:
[tex]\[
\frac{-\frac{16}{3}}{-\frac{8}{3}} = \frac{16}{8} = 2
\][/tex]
- The ratio of the third term to the second term is:
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2
\][/tex]
- The ratio of the fourth term to the third term is:
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = \frac{64}{32} = 2
\][/tex]
- The ratio of the fifth term to the fourth term is:
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = \frac{128}{64} = 2
\][/tex]
In each case, the ratio is the same, which is 2. This tells us that the sequence is geometric with a common ratio of 2.
Therefore, the formula to describe this sequence is:
[tex]\[ f(x+1) = 2 f(x) \][/tex]
This means that each term in the sequence is twice the previous term.
