Answer :
Sure! Let's analyze the sequence and how it changes from one term to the next to find a formula that describes it.
Here's the sequence you provided:
[tex]\[
-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}
\][/tex]
First, let's convert these mixed numbers into improper fractions for easier analysis:
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]
Now, let's see how the sequence progresses from one term to the next. To do this, we'll calculate the ratio between each successive pair of terms:
- From [tex]\(-\frac{8}{3}\)[/tex] to [tex]\(-\frac{16}{3}\)[/tex], the ratio is [tex]\(\frac{-16/3}{-8/3} = 2\)[/tex].
- From [tex]\(-\frac{16}{3}\)[/tex] to [tex]\(-\frac{32}{3}\)[/tex], the ratio is [tex]\(\frac{-32/3}{-16/3} = 2\)[/tex].
- From [tex]\(-\frac{32}{3}\)[/tex] to [tex]\(-\frac{64}{3}\)[/tex], the ratio is [tex]\(\frac{-64/3}{-32/3} = 2\)[/tex].
- From [tex]\(-\frac{64}{3}\)[/tex] to [tex]\(-\frac{128}{3}\)[/tex], the ratio is [tex]\(\frac{-128/3}{-64/3} = 2\)[/tex].
As we can see, the ratio between each consecutive term is consistently [tex]\(2\)[/tex]. This means each term is obtained by multiplying the previous term by [tex]\(2\)[/tex].
Thus, the formula that describes the sequence is:
[tex]\[ f(x+1) = 2f(x) \][/tex]
However, notice that this formula results in terms that are strictly positive and increasing, which is not the case in our sequence where terms are negative and the magnitude increases. Discounting the sign inconsistency, the numerical factor holds true. Therefore, the observed pattern for magnitude is correctly matched by the function [tex]\(f(x+1) = -2f(x)\)[/tex].
Here's the sequence you provided:
[tex]\[
-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}
\][/tex]
First, let's convert these mixed numbers into improper fractions for easier analysis:
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]
Now, let's see how the sequence progresses from one term to the next. To do this, we'll calculate the ratio between each successive pair of terms:
- From [tex]\(-\frac{8}{3}\)[/tex] to [tex]\(-\frac{16}{3}\)[/tex], the ratio is [tex]\(\frac{-16/3}{-8/3} = 2\)[/tex].
- From [tex]\(-\frac{16}{3}\)[/tex] to [tex]\(-\frac{32}{3}\)[/tex], the ratio is [tex]\(\frac{-32/3}{-16/3} = 2\)[/tex].
- From [tex]\(-\frac{32}{3}\)[/tex] to [tex]\(-\frac{64}{3}\)[/tex], the ratio is [tex]\(\frac{-64/3}{-32/3} = 2\)[/tex].
- From [tex]\(-\frac{64}{3}\)[/tex] to [tex]\(-\frac{128}{3}\)[/tex], the ratio is [tex]\(\frac{-128/3}{-64/3} = 2\)[/tex].
As we can see, the ratio between each consecutive term is consistently [tex]\(2\)[/tex]. This means each term is obtained by multiplying the previous term by [tex]\(2\)[/tex].
Thus, the formula that describes the sequence is:
[tex]\[ f(x+1) = 2f(x) \][/tex]
However, notice that this formula results in terms that are strictly positive and increasing, which is not the case in our sequence where terms are negative and the magnitude increases. Discounting the sign inconsistency, the numerical factor holds true. Therefore, the observed pattern for magnitude is correctly matched by the function [tex]\(f(x+1) = -2f(x)\)[/tex].
