The Correct Answer and Explanation is:
The correct answer is F. 2.
The image provided for the question is incomplete, as it refers to an “infinitely repeating decimal above” that is not shown. However, this is a classic type of mathematical problem, and the context strongly suggests that the missing decimal is the expansion of the fraction 1/7. The decimal representation of 1/7 is 0.142857142857…, where the sequence of six digits “142857” repeats infinitely. The answer choices provided are consistent with the digits found in this particular sequence.
To solve the problem, we must identify the repeating pattern and its length. The repeating block, or repetend, is “142857”. The length of this repeating block is 6 digits.
The question asks for the 381st digit after the decimal point. To find this, we can determine where the 381st position falls within the 6 digit cycle. This is done by dividing 381 by the length of the repeating block, which is 6, and finding the remainder.
The calculation is:
381 ÷ 6
Six goes into 381 a total of 63 times, with a value left over. We can calculate this as 6 × 63 = 378. The remainder is the difference between 381 and 378, which is 3.
This remainder of 3 is the key to finding the answer. It means that the 381st digit is the same as the 3rd digit in the repeating pattern. The sequence completes 63 full cycles of “142857” and then continues for three more positions into the next cycle.
Looking at the repeating block “142857”:
- The 1st digit is 1.
- The 2nd digit is 4.
- The 3rd digit is 2.
Therefore, the 381st digit after the decimal point is 2.
