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A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf ?

A  leaf  is  torn  from  a  paperback  novel. The  sum  of  the  numbers  on  the  remaining  pages  is 15000.  What are the page numbers on the torn leaf ?

Grade:10

2 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Please find the answer to your question below
The sum of the numbers form an arithmetic progression.
So, assuming the number of pages is n, the first is number 1. And number missing are p and p+1 (a leaf is torn so two page numbers missing).
Then Sum of all pages= n/2 (2+n-1) {using the formula of sum of AP with a=1}
The Sum of remaining pages =15000=n/2(2+n-1)-p-(p+1);
=> 15000= n+n^2/2-n/2-2p-1
=> 15000=n^2/2+n/2-2p-1
=>14999*2=n^2+n-4p
=>4p=n^2+n-14999*2
=>p=(n^2+n-14999*2)*0.25
=>p=(n(n+1)-14999*2)*0.25
Thus knowing the total number of pages you can determine p and p+1 numerically.
Arimaspi
19 Points
6 years ago
first, we must lay down some rules which are followed by every book
1. the total number of pages of any book will always be even
2.the page numbers of any leaf will always be an odd number followed by an even number
 
since the sum of pages after tearing one leaf is exactly 15000, the sum including the torn leaf would be greater than 15000
the least value of n which satisfies the inequality n(n+1)/2>15000 is 173
if n=173, the torn page numbers will turn out be 25 and 26. but since 173 is odd this answer is wrong.
if n=174, the pages will be 112 and 113, but these are not on the same leaf.
n=175 is automatically rejected (since 175 is odd)
n=176 will give the page numbers even greater than 176, which is the total number of pages.
 
hence the condition given in the question is never satisfied
this case is impossible and an answer does not exist

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