We appreciate your visit to Given the sequence tex 2 frac 2 3 5 frac 1 3 10 frac 2 3 21 frac 1 3 42 frac 2 3 ldots. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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].
Thanks for taking the time to read Given the sequence tex 2 frac 2 3 5 frac 1 3 10 frac 2 3 21 frac 1 3 42 frac 2 3 ldots. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
