Answer :
To determine the formula that represents the sequence, let's first understand how the terms change as we move from one term to the next.
The sequence given is:
-2⅔, -5⅓, -10⅔, -21⅓, -42⅔, ...
To make it easier to work with, let's convert each term to an improper fraction:
1. -2⅔ becomes [tex]\(-\frac{8}{3}\)[/tex]
2. -5⅓ becomes [tex]\(-\frac{16}{3}\)[/tex]
3. -10⅔ becomes [tex]\(-\frac{32}{3}\)[/tex]
4. -21⅓ becomes [tex]\(-\frac{64}{3}\)[/tex]
5. -42⅔ becomes [tex]\(-\frac{128}{3}\)[/tex]
Now, let's find the relationship between these terms by examining the ratio between consecutive terms:
[tex]\[
\text{Ratio} = \frac{\text{next term}}{\text{current term}}
\][/tex]
For example, the ratio between the first term and the second term:
[tex]\[
\text{Ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
This tells us that each term is obtained by multiplying the previous term by 2.
Therefore, the formula for this sequence is:
[tex]\[ f(x+1) = 2f(x) \][/tex]
This formula shows that each term in the sequence is twice the previous term.
The sequence given is:
-2⅔, -5⅓, -10⅔, -21⅓, -42⅔, ...
To make it easier to work with, let's convert each term to an improper fraction:
1. -2⅔ becomes [tex]\(-\frac{8}{3}\)[/tex]
2. -5⅓ becomes [tex]\(-\frac{16}{3}\)[/tex]
3. -10⅔ becomes [tex]\(-\frac{32}{3}\)[/tex]
4. -21⅓ becomes [tex]\(-\frac{64}{3}\)[/tex]
5. -42⅔ becomes [tex]\(-\frac{128}{3}\)[/tex]
Now, let's find the relationship between these terms by examining the ratio between consecutive terms:
[tex]\[
\text{Ratio} = \frac{\text{next term}}{\text{current term}}
\][/tex]
For example, the ratio between the first term and the second term:
[tex]\[
\text{Ratio} = \frac{-\frac{16}{3}}{-\frac{8}{3}} = 2
\][/tex]
This tells us that each term is obtained by multiplying the previous term by 2.
Therefore, the formula for this sequence is:
[tex]\[ f(x+1) = 2f(x) \][/tex]
This formula shows that each term in the sequence is twice the previous term.
