There is no such thing as a non-repeating division sequence if the quotient is formed by the division of two integers or by the divsion of two rational numbers.

Orokusaki, any rational sequence of any length (length=$n$) at all can be constructed as a fraction of the form

$$\frac{\sum_{i=1}^{i=n} d_i 10^{i-1}}{10^n -1}$$

where $d_i$ is the $i$-th digit counting from the right (the lowermost powers of 10).

For example, the fractional sequence repeating 12 as 0.121212... can be constructed as $12/99$.

The repeating decimal 0.121212121213 which continually repeats 12 five times followed by 13 once is the fraction

$$\frac{121212121213}{999999999999}$$

So the fact that you have $m$ repeats of a $n$-digit length sequence is not a guarantee that the repeats will continue ad infinitum.

Sequences which repeat a sequence a fixed number of times followed by infinite repeats of another sequence can also be constructed in a similar fashion.

If, however, you are generating this digit sequence as the quotient of two integer values (call them $a,b, a\in \mathbb{Z}, b \in \mathbb{Z}$), then you are guaranteed that the sequence will repeat. This is because all rational fractions of integers can be represented in the fraction format $c/({10^n-1})$ with $c \in \mathbb{Z}$ as described above, though it is not always easy to determine what length of repeat may occur.

3 spelling correction and clarification

Orokusaki, any rational sequence of any length (length=$n$) at all can be constructed as a fraction of the form

$$\frac{\sum_{i=1}^{i=n} d_i 10^{i-1}}{10^n -1}$$

where $d_i$ is the $i$-th digit counting from the right (the lowermost powers of 10).

For example, the fractional sequence repeating 12 as 0.121212... can be constructed as $12/99$.

The repeating decimal 0.121212121213 which continually repeats 12 five times followed by 13 once is the fraction

$$\frac{121212121213}{999999999999}$$

So the fact that you have $m$ repeats of a $n$-digit length sequence is not a guarantee that the repeats will continue ad infinitum.

Sequences which repeat a sequence a fixed number of times followed by infinite repeats of another sequence can also be constructed in a similar fashion.

If, however, you are generating this digit sequence as the quotient of two integer values (call then them $a,b, a\in \mathbb{Z}, b \in \mathbb{Z}$), then you are guaranteed that the sequence will repeat. This is because all rational fractions of integers can be represented in the fraction format $c/({10^n-1})$ with $c \in \mathbb{Z}$ as described above, though it is not always easy to determine what length of repeat may occur.

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Orokusaki, any rational sequence of any length (length=$n$) at all can be constructed as a fraction of the form

$$\frac{\sum_{i=1}^{i=n} d_i 10^{i-1}}{10^n -1}$$

where $d_i$ is the $i$-th digit counting from the right (the lowermost powers of 10).

For example, the fractional sequence repeating 12 as 0.121212... can be constructed as $12/99$.

The repeating decimal 0.121212121213 which continually repeats 12 five times followed by 13 once is the fraction

$$\frac{121212121213}{999999999999}$$

So the fact that you have $m$ repeats of a $n$-digit length sequence is not a guarantee that the repeats will continue ad infinitum.

Sequences which repeat a sequence a fixed number of times followed by infinite repeats of another sequence can also be constructed in a similar fashion.

If, however, you are generating this digit sequence as the quotient of two integer values (call then $a,b, a\in \mathbb{Z}, b \in \mathbb{Z}$), then you are guaranteed that the sequence will repeat. This is because all rational fractions of integers can be represented in the fraction format $c/({10^n-1})$ with $c \in \mathbb{Z}$ as described above.

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