Loading...
The string of digits \(135791113...999\) is formed by merging together the decimal representations of the odd integers from \(1\) through \(999\). Counting from left, what is the \(110^{\mathrm{th}}\) digit of this string of digits?
Let's start by understanding what we're looking for. We have a long string of digits created by writing out all the odd numbers from 1 to 999, one after another, without any spaces or separators.
So our string looks like: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33...
When we write this as one continuous string, it becomes: 135791113151719212325272931333...
We need to find what digit appears in the 110th position when counting from left to right.
Process Skill: TRANSLATE - Converting the word problem into a clear mathematical task
To solve this systematically, let's think about how many digits each type of number contributes:
1-digit odd numbers: 1, 3, 5, 7, 9
That's 5 numbers, each contributing 1 digit = 5 total digits
2-digit odd numbers: 11, 13, 15, 17, ..., 97, 99
To count these: The first 2-digit odd number is 11, the last is 99
Odd numbers from 11 to 99: We can think of this as \((99-11)/2 + 1 = 44 + 1 = 45\) numbers
Each contributes 2 digits = \(45 \times 2 = 90\) total digits
3-digit odd numbers: 101, 103, 105, ..., 997, 999
The first 3-digit odd number is 101, the last is 999
Odd numbers from 101 to 999: \((999-101)/2 + 1 = 449 + 1 = 450\) numbers
Each contributes 3 digits = \(450 \times 3 = 1,350\) total digits
Now let's track our progress toward the 110th digit:
After all 1-digit odd numbers: 5 digits total
After all 2-digit odd numbers: \(5 + 90 = 95\) digits total
After all 3-digit odd numbers: \(95 + 1,350 = 1,445\) digits total
Since we need the 110th digit, and we have 95 digits after finishing all 1-digit and 2-digit numbers, we need to look into the 3-digit numbers.
We need: \(110 - 95 = 15\) more digits into the 3-digit section.
Now we need to find which 3-digit odd number contains the 15th digit in that section.
Since each 3-digit number contributes 3 digits:
So the 15th digit in the 3-digit section is the last digit of the 5th 3-digit odd number.
The 5th 3-digit odd number is: \(101 + 2 \times 4 = 101 + 8 = 109\)
The last digit of 109 is 9.
Process Skill: APPLY CONSTRAINTS - Systematically working through the positioning to ensure accuracy
The 110th digit in the string is 9.
This matches answer choice (E) 9.
1. Misunderstanding the sequence pattern: Students may think the sequence includes ALL integers from 1 to 999, rather than recognizing it only includes ODD integers. This fundamental misreading would lead them to count even numbers like 2, 4, 6, 8, 10, 12, etc., completely throwing off their digit counting.
2. Failing to systematically categorize by digit length: Students might attempt to manually count digits or try to find patterns without organizing numbers by their digit count (1-digit, 2-digit, 3-digit). This unstructured approach makes it nearly impossible to efficiently locate the 110th position.
3. Not recognizing the need for cumulative positioning: Students may focus only on individual number positions rather than understanding they need to track cumulative digit positions across the entire sequence to determine where the 110th digit falls.
1. Incorrect counting of odd numbers in each category: Students often struggle with the formula (last - first)/2 + 1 for counting odd numbers in a range. For example, they might incorrectly count 2-digit odd numbers as 44 instead of 45, or make similar errors with 3-digit numbers.
2. Arithmetic errors in cumulative digit calculations: When calculating total digits contributed by each category (like 5×1=5 for 1-digit, 45×2=90 for 2-digit), students may make multiplication errors or incorrectly add cumulative totals (5+90=95).
3. Positioning errors within the target category: Once determining that the 110th digit falls in the 3-digit section, students may incorrectly calculate which specific 3-digit number contains that position, or miscalculate that they need the 15th digit (110-95=15) in that section.
1. Identifying the wrong digit within the target number: After correctly finding that the target number is 109, students might select the first digit (1) or middle digit (0) instead of recognizing that they need the third digit (9) of that number.
2. Off-by-one errors in final positioning: Students may correctly identify the target number as 109 but make a final positioning error, thinking they need a digit from the adjacent number (107 or 111) due to miscounting the exact position within the 3-digit sequence.