The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point;...

1. Translate the problem requirements: The decimal \(\mathrm{d = 0.123456789101112...}\) is formed by writing all positive integers consecutively after the decimal point. We need to find what digit appears in the 100th position after the decimal point.
2. Count digits contributed by single-digit numbers: Determine how many digits the numbers 1-9 contribute and how many positions this accounts for.
3. Count digits contributed by two-digit numbers: Calculate how many digits the numbers 10-99 contribute and determine if the 100th digit falls within this range.
4. Locate the specific number and digit: Once we know which range contains the 100th digit, identify the exact number and which digit within that number corresponds to position 100.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're dealing with here. We have a decimal number d that looks like this: \(\mathrm{d = 0.123456789101112...}\)

Think of it this way: imagine you're writing down all the counting numbers in order - 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13... - but instead of putting spaces between them, you just write them one after another after a decimal point.

So the decimal becomes: \(\mathrm{0.123456789101112131415...}\)

Our job is to figure out what digit appears in the 100th position after the decimal point. In other words, if we count 100 places to the right of the decimal point, what digit will we land on?

Process Skill: TRANSLATE - Converting the abstract description into a concrete counting problem

2. Count digits contributed by single-digit numbers

Let's start by figuring out how much space the single-digit numbers take up.

The single-digit positive integers are: 1, 2, 3, 4, 5, 6, 7, 8, 9

How many of these numbers are there? There are 9 single-digit numbers.

Since each single-digit number contributes exactly 1 digit to our decimal, the single-digit numbers contribute a total of 9 digits.

So after writing 1, 2, 3, 4, 5, 6, 7, 8, 9, we've used up positions 1 through 9 in our decimal.

This means we still need to find \(\mathrm{100 - 9 = 91}\) more digits to reach the 100th position.

3. Count digits contributed by two-digit numbers

Now let's look at the two-digit numbers: 10, 11, 12, 13, ..., 99

How many two-digit numbers are there? The two-digit numbers go from 10 to 99, so there are \(\mathrm{99 - 10 + 1 = 90}\) two-digit numbers.

Since each two-digit number contributes exactly 2 digits to our decimal, the two-digit numbers contribute a total of \(\mathrm{90 \times 2 = 180}\) digits.

Now, we needed 91 more digits to reach position 100, and the two-digit numbers can provide 180 digits total. Since \(\mathrm{91 < 180}\), this means the 100th digit definitely falls somewhere within the two-digit numbers.

So we know the 100th digit comes from one of the two-digit numbers (10, 11, 12, etc.), not from a three-digit number.

4. Locate the specific number and digit

We know that positions 1-9 are taken by single-digit numbers, so position 100 is actually the \(\mathrm{(100 - 9) = 91}\)st digit contributed by the two-digit numbers.

Now, since each two-digit number contributes 2 digits, we need to figure out which two-digit number contains the 91st digit.

Let's think about this step by step:

• The 1st and 2nd digits from two-digit numbers come from 10
• The 3rd and 4th digits come from 11
• The 5th and 6th digits come from 12
• And so on...

To find which number contains the 91st digit, we divide: \(\mathrm{91 \div 2 = 45.5}\)

Since we get 45.5, this means:

• The first 45 two-digit numbers (10, 11, 12, ..., 54) contribute \(\mathrm{45 \times 2 = 90}\) digits
• We need 1 more digit to reach the 91st position
• So the 91st digit is the first digit of the 46th two-digit number

What's the 46th two-digit number? The first two-digit number is 10, so the 46th one is \(\mathrm{10 + 45 = 55}\).

The number 55 has two digits: 5 and 5. Since we need the 1st digit of this number (because \(\mathrm{91 - 90 = 1}\)), our answer is 5.

Process Skill: APPLY CONSTRAINTS - Using the systematic counting to narrow down exactly where the 100th digit falls

4. Final Answer

The 100th digit of d to the right of the decimal point is 5.

Verification: Positions 1-9 are filled by digits 1,2,3,4,5,6,7,8,9. Positions 10-11 are filled by 1,0 (from number 10). Positions 12-13 are filled by 1,1 (from number 11), and so on. Following this pattern, position 100 corresponds to the first digit of the number 55, which is 5.

The answer is C. 5

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the decimal formation pattern

Students might think that the decimal is formed by placing decimal points between numbers (like 0.1.2.3.4...) rather than understanding that all digits are written in continuous succession after one decimal point. This leads to completely wrong counting strategies.

2. Confusing position counting with the actual digit values

Students may mistakenly think they need to find what number appears 100th in the sequence (which would be 100) rather than finding what digit appears in the 100th position after the decimal point. This fundamental misinterpretation changes the entire solution approach.

3. Not recognizing the need to count by digit groups

Students might attempt to list out all digits manually or try to find a direct formula without recognizing that they need to systematically count digits contributed by single-digit numbers, then two-digit numbers, etc. This leads to inefficient or impossible solution paths.

Errors while executing the approach

1. Arithmetic errors in counting digit contributions

Students often make mistakes when calculating how many digits each group contributes. For example, they might calculate that two-digit numbers (10-99) contribute \(\mathrm{89 \times 2 = 178}\) digits instead of the correct \(\mathrm{90 \times 2 = 180}\) digits, getting the count of two-digit numbers wrong.

2. Off-by-one errors when determining which number contains the target digit

When dividing \(\mathrm{91 \div 2 = 45.5}\), students might incorrectly conclude they need the 45th two-digit number instead of the 46th, or they might forget that the first two-digit number is 10, not 1, leading to identifying the wrong number (like 54 instead of 55).

3. Incorrect remainder interpretation

After determining that 45 complete two-digit numbers contribute 90 digits, students might incorrectly identify which digit of the 46th number (55) they need, perhaps choosing the second '5' instead of the first '5'.

Errors while selecting the answer

1. Selecting the wrong digit from the identified number

Even after correctly identifying that the 100th position falls within the number 55, students might select the second digit (5) instead of the first digit (5). In this case, both digits are the same so the error wouldn't show, but if it were a number like 54, they might choose 4 instead of 5.

2. Confusing the identified number with the answer

Students might correctly determine that the 100th digit comes from the number 55, but then mistakenly think the answer is 55 rather than the specific digit 5, leading them to look for 55 among the answer choices or make an incorrect selection.