Sums of Consecutive Positive IntegersDate: 03/02/2001 at 10:17:27 From: Loulie Subject: Consecutive sums I am looking at sums of consecutive numbers: i.e. 3 + 4 + 5 = 12. I have calculated formulae for these, but I am interested in the numbers that you CANNOT obtain by adding consecutive numbers. Using only two numbers you can obtain every odd number. The numbers that you cannot obtain are 2, 2^2, 2^3, 2^4, etc., i.e. 2, 4, 8, 16, etc. I have no idea why this is, though - can you please help me to understand and thus prove why powers of 2 are unobtainable? Thanks, Loulie Date: 03/02/2001 at 11:52:27 From: Doctor Greenie Subject: Re: Consecutive sums Hi, Loulie - First, let's clear up some language: The term "consecutive numbers" is meaningless, because if a and b are two different numbers, then the average of the two numbers is between them. So you aren't looking at sums of consecutive numbers, you are looking at sums of consecutive INTEGERS. ANY integer "n" can be written as the sum of consecutive integers: [-(n-1)]+[-(n-2)]+...+[-1]+0+1+2+...+(n-2)+(n-1)+n = n For example, the number 4 CAN be written as the sum of consecutive integers: (-3)+(-2)+(-1)+0+1+2+3+4 = 4 So you are not just looking at sums of consecutive integers, you are looking at sums of consecutive POSITIVE INTEGERS. And now on to an investigation as to why the powers of 2 are the only numbers you can't get as the sum of a series of consecutive positive integers.... The average of a set of numbers is defined as the sum of all the numbers divided by how many numbers there are. We can turn this definition around to say that the sum of any set of numbers is the average of the numbers multiplied by how many numbers there are. Using this "definition" of the sum of a set of numbers is useful in finding the sum of any finite arithmetic series, because in any arithmetic series it is always easy (due to the fact that the numbers in any arithmetic series are, by definition, "equally spaced") to find the number of numbers in the series and the average of the numbers in the series: average = (first + last)/2 (because the numbers are equally spaced) # of numbers = [(last - first)/common difference] + 1 We can use this "definition" of the sum of a set of numbers, because the sum of consecutive integers is an arithmetic series in which the difference between consecutive elements of the series is 1. We need to consider two cases: sums of consecutive integers with an odd number of elements, and sums with on even number of elements. (1) If the number of elements is odd, then the first and last elements are either both even or both odd, and the average of all the elements of the series is therefore an integer. So in this case the sum of the elements of the series is (number of elements) times (average of elements), which for this case is (odd integer) times (integer). (2) If the number of elements is even, then the first and last elements are one even and one odd, and the average of all the elements of the series is therefore halfway between two integers. So in this case the sum of the elements of the series is (number of elements) times (average of elements), which for this case is (even integer) times (number halfway between two integers). In this multiplication, we can divide the first factor by two and multiply the second factor by two; half of an even integer is an integer, and two times a number that is halfway between two integers is an odd integer. So in this case the sum of the series is (integer) times (odd integer). We have shown that the sum of ANY series of consecutive integers can be expressed as (integer) times (odd integer). Therefore, the only positive integers that CANNOT be expressed as the product of an integer and an odd integer are those integers whose only factors are even. Because 2 is the only even prime number, the only integers that cannot be expressed as the sum of a series of consecutive positive integers are the powers of 2. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 01/04/2002 at 17:30:21 From: Gus Rabson Subject: Consecutive sums The above is an erroneous proof. It proves that powers of 2 cannot be expressed as sums of consecutive integers. The proof is very elegant. But it does not prove that there are no other integers with that property. Gus Date: 01/06/2002 at 02:25:19 From: Doctor Greenie Subject: Re: Consecutive sums Hello, Gus - I think the proof as I presented it is complete. I showed that a number can be written as the sum of consecutive positive integers if, and only if, its prime factorization includes an odd prime factor; then, since 2 is the only even prime number, the conclusion is that the powers of 2 are the only numbers that cannot be written as the sum of consecutive positive integers. Here is another approach to the same problem.... Suppose we have a number n which we want to write as the sum of consecutive positive integers. Suppose first that the number n is a prime number. n = 2 can clearly not be written as the sum of consecutive positive integers, because the first two positive integers are 1 and 2. And all prime numbers other than 2 are odd, and any odd number of the form 2k+1 (k an integer) is the sum of the two consecutive integers k and k+1. So we have shown that the only prime number that cannot be written as the sum of consecutive positive integers is 2. Now let's consider the case where our number n is not prime. We can write the number n as n = (p)(m) where p is a prime and m may or may not be prime. In this case, if the prime factor p is not 2, then it is an odd number, of the form 2k+1. Then we have n = (2k+1)(m) = 2km + m This number is the sum of the consecutive integers (m-k), (m-k+1), ... , (m-1), m, (m+1), ... (m+k-1), (m+k) This demonstrates that ANY TIME the number n contains an odd prime factor, it can be written as the sum of consecutive positive integers - therefore, the only numbers that cannot be written as the sum of consecutive positive integers are those whose prime factorizations include no odd prime numbers; that is, the prime factorization contains only factors of 2, which means the number n is a power of 2. If you still have doubts about the completeness of the proof, please write back explaining why; I will be happy to discuss this with you further. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 01/06/2002 at 11:47:44 From: Gus Rabson Subject: Consecutive sums Thanks - this last communication was very clear. I appreciate it. |
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