To solve the problem of how many pages are torn off from a book with page numbers from 1 to 100, where the sum of the remaining pages is 4949, we can follow these steps:
### Step 1: Calculate the total sum of page numbers from 1 to 100
The sum of the first n natural numbers can be calculated using the formula:
\[
S_n = \frac{n(n + 1)}{2}
\]
For n = 100:
\[
S_{100} = \frac{100 \times 101}{2} = 5050
\]
### Step 2: Determine the sum of the torn pages
Since the sum of the remaining pages is given as 4949, we can find the sum of the torn pages by subtracting the sum of the remaining pages from the total sum:
\[
\text{Sum of torn pages} = S_{100} - \text{Sum of remaining pages} = 5050 - 4949 = 101
\]
### Step 3: Analyze the possible combinations of torn pages
We need to find combinations of page numbers that sum up to 101. Since each torn page has a unique page number, we can express this as finding distinct positive integers \( p_1, p_2, \ldots, p_r \) such that:
\[
p_1 + p_2 + \ldots + p_r = 101
\]
### Step 4: Consider the maximum number of pages that can be torn off
To maximize the number of torn pages, we can start with the smallest page numbers. The sum of the first \( r \) natural numbers is given by:
\[
\frac{r(r + 1)}{2}
\]
We need to find the largest \( r \) such that:
\[
\frac{r(r + 1)}{2} \leq 101
\]
Testing values:
- For \( r = 13 \):
\[
\frac{13 \times 14}{2} = 91 \quad (\text{valid})
\]
- For \( r = 14 \):
\[
\frac{14 \times 15}{2} = 105 \quad (\text{not valid})
\]
Thus, the maximum number of pages we can consider is 13.
### Step 5: Find the combination of pages that sum to 101
We can try combinations of the first 13 pages (1 to 13) and see if we can replace some pages to reach exactly 101. The sum of the first 13 pages is 91. We need an additional 10 to reach 101.
One possible combination is:
- Torn pages: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 (sum = 91)
- Replace page 13 with page 20 (sum = 91 - 13 + 20 = 98)
- Replace page 12 with page 21 (sum = 98 - 12 + 21 = 107, which is too high)
After testing various combinations, we find that:
- Torn pages: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 (sum = 55)
- Add page 46 (sum = 55 + 46 = 101)
### Conclusion
Thus, the number of pages torn off is 10.
