What is the 489th digit in the number 123456789101112...?

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.