Answer :
Sure! Here is a detailed, step-by-step solution for finding the next fraction in the sequence:
The given sequence is:
[tex]\[
\frac{13}{15}, \frac{11}{15}, \frac{3}{5}, \frac{7}{15}, \ldots
\][/tex]
First, let's ensure all fractions have a common denominator. The fractions [tex]\(\frac{13}{15}\)[/tex], [tex]\(\frac{11}{15}\)[/tex], and [tex]\(\frac{7}{15}\)[/tex] already have a denominator of 15. The fraction [tex]\(\frac{3}{5}\)[/tex] can be converted to have 15 as the denominator:
[tex]\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\][/tex]
Now the sequence is:
[tex]\[
\frac{13}{15}, \frac{11}{15}, \frac{9}{15}, \frac{7}{15}
\][/tex]
Next, we will look at the numerators of these fractions:
[tex]\[
13, 11, 9, 7
\][/tex]
We can observe that each consecutive numerator is decreasing by 2:
[tex]\[
13 - 11 = 2, \quad 11 - 9 = 2, \quad 9 - 7 = 2
\][/tex]
So, the next numerator can be found by subtracting 2 from the last numerator:
[tex]\[
7 - 2 = 5
\][/tex]
Since the sequence of denominators is always 15, the next fraction will have the numerator 5 and the denominator 15:
[tex]\[
\frac{5}{15}
\][/tex]
Finally, let's simplify [tex]\(\frac{5}{15}\)[/tex]:
[tex]\[
\gcd(5, 15) = 5
\][/tex]
[tex]\[
\frac{5 \div 5}{15 \div 5} = \frac{1}{3}
\][/tex]
Thus, the next fraction in the sequence is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
The given sequence is:
[tex]\[
\frac{13}{15}, \frac{11}{15}, \frac{3}{5}, \frac{7}{15}, \ldots
\][/tex]
First, let's ensure all fractions have a common denominator. The fractions [tex]\(\frac{13}{15}\)[/tex], [tex]\(\frac{11}{15}\)[/tex], and [tex]\(\frac{7}{15}\)[/tex] already have a denominator of 15. The fraction [tex]\(\frac{3}{5}\)[/tex] can be converted to have 15 as the denominator:
[tex]\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\][/tex]
Now the sequence is:
[tex]\[
\frac{13}{15}, \frac{11}{15}, \frac{9}{15}, \frac{7}{15}
\][/tex]
Next, we will look at the numerators of these fractions:
[tex]\[
13, 11, 9, 7
\][/tex]
We can observe that each consecutive numerator is decreasing by 2:
[tex]\[
13 - 11 = 2, \quad 11 - 9 = 2, \quad 9 - 7 = 2
\][/tex]
So, the next numerator can be found by subtracting 2 from the last numerator:
[tex]\[
7 - 2 = 5
\][/tex]
Since the sequence of denominators is always 15, the next fraction will have the numerator 5 and the denominator 15:
[tex]\[
\frac{5}{15}
\][/tex]
Finally, let's simplify [tex]\(\frac{5}{15}\)[/tex]:
[tex]\[
\gcd(5, 15) = 5
\][/tex]
[tex]\[
\frac{5 \div 5}{15 \div 5} = \frac{1}{3}
\][/tex]
Thus, the next fraction in the sequence is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
