To find the next three numbers in the repeating decimal produced by the fraction [tex]\(\frac{6}{7}\)[/tex], we need to first recognize the repeating sequence of the decimal representation.
1. Divide [tex]\(6\)[/tex] by [tex]\(7\)[/tex]:
[tex]\[
6 \div 7 = 0.\overline{857142}
\][/tex]
This quotient has a repeating sequence of six digits: [tex]\(857142\)[/tex].
2. Understand that the decimal repeats after every six digits:
[tex]\[
0.857142857142 \ldots
\][/tex]
The repeating sequence, as seen, is [tex]\(857142\)[/tex].
3. To find the next three digits in the sequence after listing out the repeating part a few times [tex]\( (857142857142 \ldots)\)[/tex], look at the possible places in the continuation of the sequence:
- Start by recognizing that each cycle of six digits repeats exactly.
- Consider the first 12 digits: [tex]\(857142857142\)[/tex].
4. The next three digits after these 12 digits are again part of the repeating sequence:
[tex]\[
\text{Digits 13, 14, and 15: } 857
\][/tex]
Therefore, the next three numbers after the initial repeating segment [tex]\(857142\)[/tex] are [tex]\(857\)[/tex].
