Question

What is the $4000$th digit following the decimal point in the expansion of $\frac{1}{175}$?

Answer 1

The decimal expansion of
`\((1)/(175)\) is \(0.\overline{005714}\)`. The 4000th digit falls within the repeating pattern at the 4th position, which is 1.

The fraction
`$(1)/(175)$` as a decimal is a repeating decimal with a pattern.

`\[ (1)/(175) = 0.\overline{005714} \]`

The pattern here is
`$005714$`, and it repeats infinitely.

To find the
`$4000$` th digit after the decimal point, first, determine the position of the
`$4000$` th digit within this pattern:

Divide $4000$ by the length of the repeating pattern ($6$) to find the number of complete repetitions:

`\[ 4000 / 6 = 666 \, \text{(with a remainder of 4)} \]`

This means there are 666 complete repetitions of the pattern within the first 4000 digits, with 4 additional digits.

Identify the 4th digit in the repeating pattern
`($005714$)`:

`\[ 0.005714 \]`

So, the 4000th digit falls within the 4th digit of the repeating pattern, which is 1.

Therefore, the
`$4000$` th digit following the decimal point in the expansion of
`(1)/(175)` is 1.