# Thread: The missing leaf

1. ## The missing leaf

A leaf was torn out from a book. The page numbers on the remaining pages of the book summed up to 1000. How many pages where there in this book at first? Which pages were torn out?

Thank you if you can help me with this problem.

2. Originally Posted by acc100jt
A leaf was torn out from a book. The page numbers on the remaining pages of the book summed up to 1000. How many pages where there in this book at first? Which pages were torn out?

Thank you if you can help me with this problem.
Hello,

as you may know the sum of the first natural numbers from 1 to n is calculated by:

$\displaystyle 1+2+3+...+n= \frac{1}{2} \cdot n \cdot (n+1)$

The original numbers of pages must be greater than 1000. So solve the inequality:

$\displaystyle \frac{1}{2} \cdot n \cdot (n+1) > 1000$. You'll get:

$\displaystyle n > 44.22...$. (The negative solution isn't very realistic with your problem). Because only one page is missing, the book contained 45 pages.

The sum

$\displaystyle 1+2+3+...+45=1035$. Thus the page nr. 35 is missing.

EB

3. Originally Posted by earboth
Hello,

as you may know the sum of the first natural numbers from 1 to n is calculated by:

$\displaystyle 1+2+3+...+n= \frac{1}{2} \cdot n \cdot (n+1)$

The original numbers of pages must be greater than 1000. So solve the inequality:

$\displaystyle \frac{1}{2} \cdot n \cdot (n+1) > 1000$. You'll get:

$\displaystyle n > 44.22...$. (The negative solution isn't very realistic with your problem). Because only one page is missing, the book contained 45 pages.

The sum

$\displaystyle 1+2+3+...+45=1035$. Thus the page nr. 35 is missing.

EB
You rip out a leaf you remove two pages numbered k and k+1, where k
is odd. Doing the analysis this way still gives 45 pages, but pages 17 and 18
are missing.

With this analysis 46 pages could have been posible except that pages
40 and 41 would have been missing, but the pages are not numbered that
way. (the lesser page number on a sheet should be odd).

RonL