**In the video below Charlie and Alison are trying to find ways of writing numbers as the sum of consecutive positive numbers.**

Choose some numbers of your own and try to write them as the sum of consecutive positive numbers.

**How many of the numbers from 1 to 30 can you write as the sum of consecutive numbers?**

**Here are some questions you might like to consider:**

What is special about the numbers that can be written as the sum of two consecutive numbers? Why?

What about sums of three consecutive numbers? Four consecutive numbers? Five ...?

What do you notice about the numbers you CANNOT express as the sum of consecutive numbers?

Can you prove any of your findings?

## Comments

### Any odd number can be written

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.

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### Powers of 2

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.

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### Impossible 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

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### Impossible numbers

Does everyone agree?

If so, can anyone explain why powers of $2$ $(2^0, 2^1, 2^2, 2^3, 2^4...)$ cannot be written as the sum of consecutive numbers?

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