21 Oct 12, 02:43AM
(This post was last modified: 21 Oct 12, 02:48AM by Felix-The-Ghost.)
The sum of a sequence of consecutive positive integers can be expressed as
sum = (number of elements * average of elements)
because the elements are evenly spaced numbers.
For example in the sequence [2, 3, 4, 5, 6]
5*4 = 20
In the example the sequence has an odd number of elements and thus the sum is divisible by an odd integer (in this case five)
For sequences with an even number of elements, you have a situation, where the average is not an integer, for example the sequence [2, 3, 4, 5, 6, 7, 8, 9]
8 * (5.5) = S
to make the equation easier you can divide the first factor by two and multiply the second by two because this will have the same result.
4 * 11
This makes the right factor odd, meaning the sum of such sequences are always divisible by an odd integer. Therefore the only positive numbers not expressible by such sequences are those that have only even factors. Two is the only even prime number, meaning powers of it are not divisible by odd integers, meaning that number cannot be expressed as the sum of sequential numbers.
sum = (number of elements * average of elements)
because the elements are evenly spaced numbers.
For example in the sequence [2, 3, 4, 5, 6]
5*4 = 20
In the example the sequence has an odd number of elements and thus the sum is divisible by an odd integer (in this case five)
For sequences with an even number of elements, you have a situation, where the average is not an integer, for example the sequence [2, 3, 4, 5, 6, 7, 8, 9]
8 * (5.5) = S
to make the equation easier you can divide the first factor by two and multiply the second by two because this will have the same result.
4 * 11
This makes the right factor odd, meaning the sum of such sequences are always divisible by an odd integer. Therefore the only positive numbers not expressible by such sequences are those that have only even factors. Two is the only even prime number, meaning powers of it are not divisible by odd integers, meaning that number cannot be expressed as the sum of sequential numbers.