To find the 288th digit of the sequence 1222333334444444555555555...12121212131313..., we need to follow these steps:

### 1. Break Down the Problem
We can observe that the sequence is constructed from the following parts:
- Single digits: 1, 2,...
- Groups of consecutive identical digits: 2 appears twice, 3 appears three times, 4 appears four times, and so on.
- The pattern continues with more 1s, 2s, and 3s after reaching the larger groups.

### 2. Relevant Concepts
The series can be broken down into two main components:
- The sequence of single digits.
- The sequence of repeated digits, where each integer $ n $ appears $ n $ times.

### 3. Analysis and Detail
1. **Count the digits contributed by each group**:
- 1 appears 1 time → 1 digit
- 2 appears 2 times → 2 digits
- 3 appears 3 times → 3 digits
- 4 appears 4 times → 4 digits
- ...
- $ n $ appears $ n $ times → $ n $ digits

2. **Total digits contributed by first $ n $ groups**:
To find the total number of digits contributed by the first $ n $ numbers, we sum the first $ n $ integers:
$$
S_n = 1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2}
$$

3. **Finding $ n $ such that $ S_n $ is at least 288**:
We need to determine the largest $ n $ such that:
$$
\frac{n(n + 1)}{2}

Question

To find the 288th digit of the sequence 1222333334444444555555555...12121212131313..., we need to follow these steps:

### 1. Break Down the Problem
We can observe that the sequence is constructed from the following parts:
- Single digits: 1, 2,...
- Groups of consecutive identical digits: 2 appears twice, 3 appears three times, 4 appears four times, and so on.
- The pattern continues with more 1s, 2s, and 3s after reaching the larger groups.

### 2. Relevant Concepts
The series can be broken down into two main components:
- The sequence of single digits.
- The sequence of repeated digits, where each integer $ n $ appears $ n $ times.

### 3. Analysis and Detail
1. **Count the digits contributed by each group**:
   - 1 appears 1 time → 1 digit
   - 2 appears 2 times → 2 digits
   - 3 appears 3 times → 3 digits
   - 4 appears 4 times → 4 digits
   - ...
   - $ n $ appears $ n $ times → $ n $ digits

2. **Total digits contributed by first $ n $ groups**:
   To find the total number of digits contributed by the first $ n $ numbers, we sum the first $ n $ integers:
   $$
   S_n = 1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2}
   $$

3. **Finding $ n $ such that $ S_n $ is at least 288**:
   We need to determine the largest $ n $ such that:
   $$
   \frac{n(n + 1)}{2} < 288
   $$
   This translates to solving:
   $$
   n(n + 1) < 576
   $$
   Let's approximate $ n $:
   - Testing $ n = 23 $:
     $$
     23 \cdot 24 = 552 < 576 \quad (\text{valid})
     $$
   - Testing $ n = 24 $:
     $$
     24 \cdot 25 = 600 \quad (\text{too high})
     $$
   Hence, $ n = 23 $.

4. **Confirm digits up to $ n = 23 $**:
   $$
   S_{23} = \frac{23 \cdot 24}{2} = 276
   $$
   Thus, the first 276 digits are used. The first number contributing is:
   - 1 contributes 1 digit: 1
   - 2 contributes 2 digits: 2, 2
   - 3 contributes 3 digits: 3, 3, 3
   - ...
   - 23 contributes 23 digits: 23 (23 times)

5. **Digits from $ n = 24 $**:
   Since we already have 276 digits, we need the next digits from 24:
   - 24 will contribute 24 digits: 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24.
   - The 277th digit is the first '2' in '24', the 278th digit is the first '4', continuing this way.

### 4. Verify and Summarize
The digits continue:
- 277th → '2'
- 278th → '4'
- 279th → '2'
- 280th → '4'
- 281st → '2'
- 282nd → '4'
- 283rd → '2'
- 284th → '4'
- 285th → '2'
- 286th → '4'
- 287th → '2'
- 288th → '4'

### Final Answer
The 288th digit of the sequence is **4**.

Knowee AI · Accepted Answer