Answer :
To find the next fraction in the sequence [tex]\(\frac{14}{15}, \frac{4}{5}, \frac{2}{3}, \frac{8}{15}, \ldots\)[/tex], we need to search for a pattern in the numerators of the fractions. Here's a step-by-step approach:
1. Convert each fraction to have a common denominator: Here, the common denominator is 15.
- The first fraction is already [tex]\(\frac{14}{15}\)[/tex].
- The second fraction can be rewritten as [tex]\(\frac{4}{5} = \frac{12}{15}\)[/tex].
- The third fraction can be rewritten as [tex]\(\frac{2}{3} = \frac{10}{15}\)[/tex].
- The fourth fraction is already [tex]\(\frac{8}{15}\)[/tex].
So the sequence with a common denominator of 15 is:
[tex]\[
\frac{14}{15}, \frac{12}{15}, \frac{10}{15}, \frac{8}{15}
\][/tex]
2. Identify the pattern in the numerators:
- The numerators are: 14, 12, 10, 8.
- Observe that each numerator is decreasing by 2:
[tex]\[
14 - 12 = 2, \quad 12 - 10 = 2, \quad 10 - 8 = 2
\][/tex]
3. Continue the pattern:
- The next numerator in the sequence would be [tex]\(8 - 2 = 6\)[/tex].
4. Write the next fraction with the common denominator:
- The next fraction is [tex]\(\frac{6}{15}\)[/tex].
5. Simplify the fraction:
- The greatest common divisor (GCD) of 6 and 15 is 3.
- Dividing the numerator and the denominator by 3:
[tex]\[
\frac{6 \div 3}{15 \div 3} = \frac{2}{5}
\][/tex]
Thus, the next fraction in the sequence is [tex]\(\boxed{\frac{2}{5}}\)[/tex].
1. Convert each fraction to have a common denominator: Here, the common denominator is 15.
- The first fraction is already [tex]\(\frac{14}{15}\)[/tex].
- The second fraction can be rewritten as [tex]\(\frac{4}{5} = \frac{12}{15}\)[/tex].
- The third fraction can be rewritten as [tex]\(\frac{2}{3} = \frac{10}{15}\)[/tex].
- The fourth fraction is already [tex]\(\frac{8}{15}\)[/tex].
So the sequence with a common denominator of 15 is:
[tex]\[
\frac{14}{15}, \frac{12}{15}, \frac{10}{15}, \frac{8}{15}
\][/tex]
2. Identify the pattern in the numerators:
- The numerators are: 14, 12, 10, 8.
- Observe that each numerator is decreasing by 2:
[tex]\[
14 - 12 = 2, \quad 12 - 10 = 2, \quad 10 - 8 = 2
\][/tex]
3. Continue the pattern:
- The next numerator in the sequence would be [tex]\(8 - 2 = 6\)[/tex].
4. Write the next fraction with the common denominator:
- The next fraction is [tex]\(\frac{6}{15}\)[/tex].
5. Simplify the fraction:
- The greatest common divisor (GCD) of 6 and 15 is 3.
- Dividing the numerator and the denominator by 3:
[tex]\[
\frac{6 \div 3}{15 \div 3} = \frac{2}{5}
\][/tex]
Thus, the next fraction in the sequence is [tex]\(\boxed{\frac{2}{5}}\)[/tex].
