To determine the 275th digit after the decimal point in the repeating decimal [tex]\(0.\overline{6295}\)[/tex], we need to consider the periodic nature of the decimal. The repeating sequence here is "6295", which has a length of 4 digits.
Here is the step-by-step method to find the desired digit:
1. Understand the repeating cycle:
The sequence "6295" repeats indefinitely. This means that every 4 digits, the sequence restarts from the beginning.
2. Determine the effective position within one repeating cycle:
To find which digit corresponds to the 275th position, we need to figure out where the 275th position falls within the cycle of 4 digits. This can be determined by finding the remainder when 274 is divided by 4. We use 274 because the counting starts from 1, so we adjust by subtracting 1 from the desired position.
[tex]\[
\text{Effective position} = (275 - 1) \mod 4 = 274 \mod 4
\][/tex]
3. Calculate the remainder:
When we divide 274 by 4, we get:
[tex]\[
274 \div 4 = 68 \text{ R } 2
\][/tex]
The remainder is 2, which means the 275th digit is the same as the 2nd digit in the repeating sequence.
4. Identify the digit in the sequence:
The repeating sequence "6295" is indexed as follows:
- 1st digit: 6
- 2nd digit: 2
- 3rd digit: 9
- 4th digit: 5
Therefore, the 2nd digit in this sequence is 9.
Thus, the 275th digit after the decimal point in the repeating decimal [tex]\(0.\overline{6295}\)[/tex] is:
[tex]\[
\boxed{9}
\][/tex]
So the correct answer is K.
