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Sums of Consecutive Integers


Date: 02/04/2001 at 14:04:40
From: Kasey
Subject: Sum of a string of consecutive positive numbers

Some numbers can be expressed as the sum of a string of consecutive 
positive numbers. Exactly which numbers have this property? Be 
prepared to support your decision with documentation.

Any help you can give me I would greatly appreciate. 
Thank you.


Date: 02/04/2001 at 15:00:31
From: Doctor Greenie
Subject: Re: Sum of a string of consecutive positive numbers

Hi, Kasey -

First, let's clarify some language so we are sure what we are talking 
about.

The phrase "consecutive positive numbers" is meaningless, because 
given any two positive numbers I can always find a number between 
them. I think that you really mean to be talking about consecutive 
positive integers.

It turns out there is a very small and well-defined set of positive 
integers that CANNOT be expressed as the sum of a string of 
consecutive positive integers. They are the only integers with this 
property. I won't tell you what the set is, but I will give you some 
hints to let you try to discover it yourself.

The key to discovering the answer is in how you think of finding the 
sum of a set of consecutive positive integers. There are various 
equivalent versions of the formula for finding such a sum; the one I 
always find most useful is the following:

     sum = (average) * (how many)

This is a universal truth for ANY set of numbers; it follows from the 
definition of average: since the average of any set of numbers is the 
sum of the numbers divided by how many there are, it follows that the 
sum of any set of numbers is the average of the numbers multiplied by 
how many numbers there are.

This is a very useful method for finding the sum of a set of 
consecutive positive integers, because it is always easy to find the 
number of numbers in the set and the average of those numbers.

In any arithmetic series (a series in which the difference between 
successive terms is always the same), the average of the terms in the 
series is the average of the first and last terms in the series. 
(This follows from the fact that the elements of the series are 
"evenly spaced" throughout.)

Any set of consecutive positive integers is an example of an 
arithmetic series, because the difference between successive terms is 
always 1. Such a set can have either an odd or an even number of 
terms.

If the series of consecutive positive integers has an odd number of 
terms, then the average of the numbers is an integer; in this case 
the sum of the integers is then

     (1)  sum = (average) * (how many) = (integer) * (odd integer)

If the series of consecutive positive integers has an even number of 
terms, then the average of the numbers is a number halfway between 
two integers. "A number halfway between two integers" can be written 
as "half of an odd integer," so in this case the sum of the integers 
is then

     (2)  sum = (average) * (how many) 
              = (half an odd integer) * (even integer)
              = (odd integer) * (integer)

In writing the last line above, I have modified the multiplication 
problem by doubling the first factor (two times half an odd integer 
is an odd integer) and halving the second factor (half of an even 
integer is an integer, which may be either odd or even).

Together, equations (1) and (2) above tell me that the sum of ANY 
series of consecutive positive integers can be written as the product 
of an odd integer and some other integer that may be either odd or 
even.

An analysis of this fact can lead to the determination of exactly 
which positive integers CANNOT be expressed as the sum of a series of 
consecutive positive integers.

Good luck with finishing this. Write back if you need more help.

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

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