Math Problem Statement
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
Solution
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 pages, the sum of all page numbers is given by the formula for the sum of an arithmetic series:
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 . Therefore, the equation becomes:
Step 3: Express and simplify
Rearrange the equation to solve for :
Step 4: Estimate using the quadratic formula
The sum before tearing out the page should be close to 1783 plus a reasonable page number. We solve:
Since is a page number, we estimate using the quadratic approximation:
We can approximate by solving the quadratic equation:
Using the quadratic formula:
Since must be positive:
Step 5: Check
Now, calculate the sum of the pages for :
Since the sum is not exactly 1783, try :
Now, find :
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:
- How can the sum formula for an arithmetic series be derived?
- What steps are involved in solving quadratic equations using the quadratic formula?
- Why is it necessary to check both possible values of before confirming the solution?
- What is the significance of approximating in this context?
- 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.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Series
Quadratic Equations
Formulas
Sum of an arithmetic series
Quadratic formula
Theorems
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Suitable Grade Level
Grades 9-12
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