Context: From a book having page numbers 1 to n (both sides), a page is torn out. If the sum of the page numbers of the book is 1783 now, then find the smaller of the page number of the page which was torn out

Answer question

To solve the problem, we need to determine the original number of pages in the book and identify the page number that was torn out based on the given information.

### Step 1: Find the sum of the original page numbers

If the book has \( n \) pages, the sum of all page numbers is given by the formula for the sum of an arithmetic series:

\[
\text{Sum of pages} = \frac{n(n+1)}{2}
\]

### Step 2: Set up the equation

We know that after tearing out one page, the sum of the remaining pages is 1783. Let the page that was torn out be \( x \). Therefore, the equation becomes:

\[
\frac{n(n+1)}{2} - x = 1783
\]

### Step 3: Express \( x \) and simplify

Rearrange the equation to solve for \( x \):

\[
x = \frac{n(n+1)}{2} - 1783
\]

### Step 4: Estimate \( n \) using the quadratic formula

The sum before tearing out the page should be close to 1783 plus a reasonable page number. We solve:

\[
\frac{n(n+1)}{2} \approx 1783 + x
\]

Since \( x \) is a page number, we estimate \( n \) using the quadratic approximation:

\[
n(n+1) \approx 2 \times 1783 \approx 3566
\]

We can approximate \( n \) by solving the quadratic equation:

\[
n^2 + n - 3566 = 0
\]

Using the quadratic formula:

\[
n = \frac{-1 \pm \sqrt{1 + 4 \times 3566}}{2}
\]

\[
n = \frac{-1 \pm \sqrt{14265}}{2}
\]

\[
n \approx \frac{-1 \pm 119.4}{2}
\]

Since \( n \) must be positive:

\[
n \approx \frac{118.4}{2} \approx 59.2 \approx 59
\]

### Step 5: Check \( n = 59 \)

Now, calculate the sum of the pages for \( n = 59 \):

\[
\text{Sum of pages} = \frac{59 \times 60}{2} = 1770
\]

Since the sum is not exactly 1783, try \( n = 60 \):

\[
\text{Sum of pages} = \frac{60 \times 61}{2} = 1830
\]

Now, find \( x \):

\[
x = 1830 - 1783 = 47
\]

So, the page that was torn out is 47.

### Final Answer:

The smaller of the two page numbers of the torn-out page is **47**.

Would you like more details or have any questions? 

Here are 5 related questions to further your understanding:
1. How can the sum formula for an arithmetic series be derived?
2. What steps are involved in solving quadratic equations using the quadratic formula?
3. Why is it necessary to check both possible values of \( n \) before confirming the solution?
4. What is the significance of approximating \( n \) in this context?
5. How would the problem change if a different page number was torn out?

**Tip:** When solving quadratic equations, always remember to check both potential solutions to ensure you select the correct one for the problem context.

Learn how to find the missing page number torn out from a book with pages numbered 1 to n, given that the sum of the remaining pages is 1783. This problem involves arithmetic series and quadratic equations. Suitable for high school students studying algebra and arithmetic series.

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