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Answer :
To determine the formula that describes the sequence, let's first understand the numbers given:
The sequence is:
[tex]\[
-5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots
\][/tex]
First, let's convert these mixed numbers into improper fractions:
1. [tex]\(-5 \frac{1}{3} = -(5 + \frac{1}{3}) = -\frac{16}{3}\)[/tex]
2. [tex]\(-10 \frac{2}{3} = -(10 + \frac{2}{3}) = -\frac{32}{3}\)[/tex]
3. [tex]\(-21 \frac{1}{3} = -(21 + \frac{1}{3}) = -\frac{64}{3}\)[/tex]
4. [tex]\(-42 \frac{2}{3} = -(42 + \frac{2}{3}) = -\frac{128}{3}\)[/tex]
Next, we find the common ratio between consecutive terms by dividing each term by the term that comes before it:
1. [tex]\(\text{Ratio between first and second term:}\)[/tex]
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2
\][/tex]
2. [tex]\(\text{Ratio between second and third term:}\)[/tex]
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = \frac{64}{32} = 2
\][/tex]
3. [tex]\(\text{Ratio between third and fourth term:}\)[/tex]
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = \frac{128}{64} = 2
\][/tex]
Since the ratio between each consecutive term is consistently 2, we can conclude that the sequence is geometric and each term is obtained by multiplying the previous term by 2.
Thus, the correct formula to describe this sequence is:
[tex]\[
f(x+1) = 2f(x)
\][/tex]
The sequence is:
[tex]\[
-5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots
\][/tex]
First, let's convert these mixed numbers into improper fractions:
1. [tex]\(-5 \frac{1}{3} = -(5 + \frac{1}{3}) = -\frac{16}{3}\)[/tex]
2. [tex]\(-10 \frac{2}{3} = -(10 + \frac{2}{3}) = -\frac{32}{3}\)[/tex]
3. [tex]\(-21 \frac{1}{3} = -(21 + \frac{1}{3}) = -\frac{64}{3}\)[/tex]
4. [tex]\(-42 \frac{2}{3} = -(42 + \frac{2}{3}) = -\frac{128}{3}\)[/tex]
Next, we find the common ratio between consecutive terms by dividing each term by the term that comes before it:
1. [tex]\(\text{Ratio between first and second term:}\)[/tex]
[tex]\[
\frac{-\frac{32}{3}}{-\frac{16}{3}} = \frac{32}{16} = 2
\][/tex]
2. [tex]\(\text{Ratio between second and third term:}\)[/tex]
[tex]\[
\frac{-\frac{64}{3}}{-\frac{32}{3}} = \frac{64}{32} = 2
\][/tex]
3. [tex]\(\text{Ratio between third and fourth term:}\)[/tex]
[tex]\[
\frac{-\frac{128}{3}}{-\frac{64}{3}} = \frac{128}{64} = 2
\][/tex]
Since the ratio between each consecutive term is consistently 2, we can conclude that the sequence is geometric and each term is obtained by multiplying the previous term by 2.
Thus, the correct formula to describe this sequence is:
[tex]\[
f(x+1) = 2f(x)
\][/tex]
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