Answer :
Let's take a look at the sequence of fractions: [tex]\(\frac{14}{15}, \frac{4}{5}, \frac{2}{3}, \frac{8}{15}, \ldots\)[/tex]. We need to find the next fraction in this sequence.
First, let’s convert these fractions to decimal form to analyze them more easily:
1. [tex]\(\frac{14}{15} \approx 0.9333\)[/tex]
2. [tex]\(\frac{4}{5} = 0.8\)[/tex]
3. [tex]\(\frac{2}{3} \approx 0.6667\)[/tex]
4. [tex]\(\frac{8}{15} \approx 0.5333\)[/tex]
Next, we'll check the difference between the consecutive terms to find a pattern:
- The difference between the first and second term: [tex]\(0.8 - 0.9333 = -0.1333\)[/tex]
- The difference between the second and third term: [tex]\(0.6667 - 0.8 = -0.1333\)[/tex]
- The difference between the third and fourth term: [tex]\(0.5333 - 0.6667 = -0.1333\)[/tex]
We can see that each term is obtained by subtracting approximately [tex]\(0.1333\)[/tex] from the previous term.
To find the next term in the sequence, we can subtract [tex]\(0.1333\)[/tex] from the last known term, [tex]\(\frac{8}{15}\)[/tex]:
- Next term in decimal form: [tex]\(0.5333 - 0.1333 = 0.4\)[/tex]
Now, we need to convert this decimal back into a simplified fraction. The decimal [tex]\(0.4\)[/tex] can be written as the fraction [tex]\(\frac{2}{5}\)[/tex].
So, the next fraction in the sequence is [tex]\(\frac{2}{5}\)[/tex].
First, let’s convert these fractions to decimal form to analyze them more easily:
1. [tex]\(\frac{14}{15} \approx 0.9333\)[/tex]
2. [tex]\(\frac{4}{5} = 0.8\)[/tex]
3. [tex]\(\frac{2}{3} \approx 0.6667\)[/tex]
4. [tex]\(\frac{8}{15} \approx 0.5333\)[/tex]
Next, we'll check the difference between the consecutive terms to find a pattern:
- The difference between the first and second term: [tex]\(0.8 - 0.9333 = -0.1333\)[/tex]
- The difference between the second and third term: [tex]\(0.6667 - 0.8 = -0.1333\)[/tex]
- The difference between the third and fourth term: [tex]\(0.5333 - 0.6667 = -0.1333\)[/tex]
We can see that each term is obtained by subtracting approximately [tex]\(0.1333\)[/tex] from the previous term.
To find the next term in the sequence, we can subtract [tex]\(0.1333\)[/tex] from the last known term, [tex]\(\frac{8}{15}\)[/tex]:
- Next term in decimal form: [tex]\(0.5333 - 0.1333 = 0.4\)[/tex]
Now, we need to convert this decimal back into a simplified fraction. The decimal [tex]\(0.4\)[/tex] can be written as the fraction [tex]\(\frac{2}{5}\)[/tex].
So, the next fraction in the sequence is [tex]\(\frac{2}{5}\)[/tex].
