Summing Consecutive Numbers

Any odd number can be written as the sum of 2 consecutive numbers, you just half it then use the number 0.5 above and 0.5 below it.
E.G. 33
33/2 = 16.5
33 = 16+17.

Any number can be written as the sum of consecutive positive numbers as long as it always has an odd factor (thus this excludes powers of two). This is because a number is the sum of consecutive numbers if the 'arrangement's' median is a factor of the number. However, if we divide, for example, 28 by 4 then we get 7. The arrangement is then going to have 4 whole numbers, where, for it to work, 7 must be in the very centre, which is impossible as 4 is even. There can be arrangements of 4 numbers (or other powers of 2), but as long as the number we are trying to total is not a multiple of one of these. That is not to say that all multiples of 4 won't work as 28 can be arranged in as the sum of 7 numbers: 1 + 2 + 3 + 4 + 5 + 6 + 7.

So basically if a number has no odd factors then it can not be represented as the sum of consecutive numbers. There is, of course, another question of whether these numbers are infinite in quantity, and if it goes for any power of two. Obviously, the powers of two are infinite and as their prime decomposition will be 2 to some power then we can see that any factor must therefore be a power of 2 (being made up of the original number's prime factors only). So in fact any power of two can not be made as the sum of consecutive numbers.

We found out that you cannot do 1, 2, 4, 8 or 16 in our investigation and we also worked out that this is because if you double 1, you get 2 and if you double 2 you get 4, etc.

We predict that you won't be able to find the numbers 32, 64, 128 and 256 because you keep doubling the number.

Eli, James, Charlie, George - Year 3, St Francis CE Primary School