Answer :
Sure, let's work through this step-by-step to find the next fraction in the sequence [tex]\(\frac{14}{15}, \frac{13}{15}, \frac{4}{5}, \frac{11}{15}, \ldots\)[/tex].
1. Observe the pattern:
- The first term is [tex]\(\frac{14}{15}\)[/tex].
- The second term is [tex]\(\frac{13}{15}\)[/tex], which is the first term's numerator decreased by 1.
- The third term is [tex]\(\frac{4}{5}\)[/tex].
Let’s simplify [tex]\(\frac{4}{5}\)[/tex] to have the same denominator, [tex]\(15\)[/tex]:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
- The fourth term is [tex]\(\frac{11}{15}\)[/tex], which continues the pattern of decreasing the numerator by 1.
2. Continue the pattern:
- The numerators are decreasing: [tex]\(14, 13, 12, 11\)[/tex].
- Thus, the next numerator should be [tex]\(10\)[/tex].
3. Construct the next fraction:
- With the next numerator being [tex]\(10\)[/tex] and the consistent denominator [tex]\(15\)[/tex], the next fraction in the sequence is:
[tex]\[
\frac{10}{15}
\][/tex]
4. Simplify:
[tex]\[
\frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3}
\][/tex]
Therefore, the next fraction in the sequence is [tex]\(\boxed{\frac{2}{3}}\)[/tex].
1. Observe the pattern:
- The first term is [tex]\(\frac{14}{15}\)[/tex].
- The second term is [tex]\(\frac{13}{15}\)[/tex], which is the first term's numerator decreased by 1.
- The third term is [tex]\(\frac{4}{5}\)[/tex].
Let’s simplify [tex]\(\frac{4}{5}\)[/tex] to have the same denominator, [tex]\(15\)[/tex]:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
- The fourth term is [tex]\(\frac{11}{15}\)[/tex], which continues the pattern of decreasing the numerator by 1.
2. Continue the pattern:
- The numerators are decreasing: [tex]\(14, 13, 12, 11\)[/tex].
- Thus, the next numerator should be [tex]\(10\)[/tex].
3. Construct the next fraction:
- With the next numerator being [tex]\(10\)[/tex] and the consistent denominator [tex]\(15\)[/tex], the next fraction in the sequence is:
[tex]\[
\frac{10}{15}
\][/tex]
4. Simplify:
[tex]\[
\frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3}
\][/tex]
Therefore, the next fraction in the sequence is [tex]\(\boxed{\frac{2}{3}}\)[/tex].
