Sachin writes all the numbers from 1 to 1000 on a paper in order. Find the 2883rd digit
written by him.

To solve the problem, we need to determine the 2883rd digit written by Sachin when he writes all the numbers from 1 to 1000 in order.

### Step 1: Count the digits used for numbers 1 to 9
- Numbers from 1 to 9 are single-digit numbers.
- There are 9 numbers, so the total digits used = \( 9 \times 1 = 9 \) digits.

### Step 2: Count the digits used for numbers 10 to 99
- Numbers from 10 to 99 are two-digit numbers.
- There are \( 99 - 10 + 1 = 90 \) numbers.
- The total digits used = \( 90 \times 2 = 180 \) digits.

### Step 3: Count the digits used for numbers 100 to 999
- Numbers from 100 to 999 are three-digit numbers.
- There are \( 999 - 100 + 1 = 900 \) numbers.
- The total digits used = \( 900 \times 3 = 2700 \) digits.

### Step 4: Count the digits used for the number 1000
- The number 1000 has 4 digits.
- Total digits used = 4 digits.

### Step 5: Calculate the cumulative number of digits
- Digits from 1 to 9: 9 digits
- Digits from 10 to 99: 180 digits
- Digits from 100 to 999: 2700 digits
- Digits from 1000: 4 digits

The cumulative number of digits = \( 9 + 180 + 2700 + 4 = 2893 \) digits.

### Step 6: Determine the 2883rd digit
- Digits from 1 to 99: \( 9 + 180 = 189 \) digits.
- The 2883rd digit is within the range of digits from 100 to 999, since up to 99 uses only 189 digits, and digits from 100 onward start from the 190th digit.
  
The position of the 2883rd digit relative to the digits from 100 to 999 is \( 2883 - 189 = 2694 \).

Now, we need to find the exact number:
- Each number from 100 to 999 uses 3 digits.
- So, the 2694th digit corresponds to the \( \left\lceil \frac{2694}{3} \right\rceil = 898 \)th number in this range.
- The 898th number from 100 is \( 100 + 897 = 997 \).

### Conclusion:
- The first digit of 997 is 9.
- The second digit is 9.
- The third digit is 7.

Thus, the 2694th digit corresponds to the third digit of 997, which is **7**. Therefore, the 2883rd digit written by Sachin is **7**.

Would you like further details or explanations on any part of the process? Here are some additional related questions:

1. How many digits would be written if Sachin wrote numbers from 1 to 2000?
2. What is the 500th digit written by Sachin?
3. What is the last digit of the sequence when Sachin writes numbers from 1 to 1000?
4. How many times does the digit '1' appear from 1 to 1000?
5. What is the sum of all digits written by Sachin from 1 to 1000?

**Tip:** When solving problems involving large sequences of numbers, breaking the sequence into smaller, manageable parts often simplifies the calculations.

Discover how to find the 2883rd digit written by Sachin when he writes numbers from 1 to 1000. This problem involves understanding digit counting, arithmetic sequences, and determining specific digits in a large sequence. Suitable for Grades 7-9 students.

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