Correct Option (4)
To determine the 489th digit in the sequence 123456789101112..., the total number of digits contributed by numbers of increasing length must be calculated sequentially.
- 1-digit numbers (1-9): There are 9 such numbers. Each contributes 1 digit.
- Total digits from 1-digit numbers = 9 × 1 = 9 digits.
- These digits occupy positions 1 through 9.
- 2-digit numbers (10-99): There are 99 - 10 + 1 = 90 such numbers. Each contributes 2 digits.
- Total digits from 2-digit numbers = 90 × 2 = 180 digits.
- Cumulative digits up to 2-digit numbers = 9 (from 1-digit) + 180 (from 2-digit) = 189 digits.
- These digits occupy positions 10 through 189.
- 3-digit numbers (100-999): The 489th digit falls within the block of 3-digit numbers, as 189 < 489.
- Digits remaining to reach the 489th position = 489 - 189 = 300 digits.
- Since each 3-digit number contributes 3 digits, the number of 3-digit numbers required to account for these 300 digits is 300 / 3 = 100 numbers.
- This implies that the 489th digit is the last digit of the 100th 3-digit number.
- The first 3-digit number is 100. The 100th 3-digit number is 100 + (100 - 1) = 199.
- The number is 199. Its digits are 1, 9, 9. The last digit of 199 is 9.
Therefore, the 489th digit in the sequence is 9.
Incorrect Options:
- (1) 0: The digit 0 would appear as part of numbers like 10, 20, 100, etc. Based on the calculation, the 489th digit is the last digit of 199, which is not 0.
- (2) 3: The digit 3 appears in various positions, but the systematic calculation places the 489th digit as 9, not 3.
- (3) 6: Similar to other incorrect options, the digit 6 does not correspond to the 489th position when the sequence is constructed and analyzed digit by digit.
