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Pages in a Book


Date: 09/13/2001 at 14:12:08
From: Patricia Anne
Subject: Unsure

A book is made of folded sheets of paper, each comprising four pages.  
One of the sheets has page numbers 88 and 169.

1. How many pages are there in the book?

2. What is the sum of all of the page numbers in the book?

3. A book has 600 pages. What is the sum of all of the page numbers in 
   the book?

Thank you for you help.


Date: 09/13/2001 at 15:21:09
From: Doctor Ian
Subject: Re: Unsure

Hi Patricia,

Let's start by looking at some very small books. Suppose the book has 
only one sheet of paper. We might indicate it this way:

  (1,2) | (3,4)

The '|' is the center of the book, and '(a,b)' indicates a pair of 
page numbers.  The next smallest book would have 8 pages:

  (1,2) (3,4) | (5,6) (7,8)

The next smallest, 12 pages:

  (1,2) (3,4) (5,6) | (7,8) (9,10) (11,12)

And so on.  

Now, here is a curious thing. Suppose the pages are labeled 1 to N.  
If we pair up pages, starting from the ends, and add each pair, what 
happens? 

      _____
     / ___ \
    / /   \ \
  (1,2) | (3,4)            each pair (1+4, 2+3) adds up to 5.

        _____________
       / ___________ \
      / /   _____   \ \
     / /   / ___ \   \ \
    / /   / /   \ \   \ \  
  (1,2) (3,4) | (5,6) (7,8)    each pair adds up to 9.

You have a sheet of paper that looks like this:

         ________________
        /  ___________   \
       /  /           \   \
    (87,88)           (169,170)

Do you see how you can use this collection of page numbers to deduce 
the number of pages in the book? 

Note that I can quickly add up the page numbers in a book by adding 
the page numbers on any sheet, and multiplying by the number of 
sheets. Do you see why this is true? 


In each of the little books we constructed, if you add up any of the 
paired page numbers, you get the number of pages in the book, plus 1. 
Since those books are constructed in the same way as yours, the same 
trick will work for your book. That is, 

  87 + 170 = 88 + 169 = the number of pages in the book + 1

Now, note that if the book has N pages, then it has N/4 sheets of 
paper. And each of those sheets of paper will contain two sets of 
pages, where each set adds up to the N+1.

So the sum of the page numbers must be

  sum of page numbers = number of sheets * 2 * (N+1)

                      = (N/4) * 2 * (N+1)


This problem is supposed to be a 'practical' application of a certain 
trick, which is useful for adding up the first N integers.  

Note that we can write the sum of N integers this way:

  1 + 2 + ... + (N-1) + N = ?

Since order is irrelevant in addition, we can also write it this way:

  N + (N-1) + ... + 2 + 1 = ?

Now, if we add corresponding elements of these sequences, we get

    1 +   2   + ... + (N-1) + N  =  ?

    N + (N-1) + ... +   2   + 1  =  ?
   ---  -----          ---   ---   ---
   N+1   N+1           N+1   N+1   2*?
   \___________________________/
             N times

So the sum must be half of 2*?, or

      N*(N+1)
  ? = -------
        2

Now, the pages in a book make up this kind of sequence. Furthermore, 
if the book is made by folding pages, as in this problem, the pages 
that appear on a single sheet correspond to pairs in the bottom 
sequence above. That is, 

  page 1   is paired with page  N
  page 2   is paired with page  N-1
  page 3   is paired with page  N-2

and so on.  So if you know that page 88 is paired with page 169, that 
is the same as pairing 87 with 170, 86 with 171, ..., all the way down 
to pairing 1 with 256. Which means that the final page must be page 
256.  And you can use the formula above to add up the pages. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Sequences, Series

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