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Summing Consecutive Numbers

Age 11 to 14
Challenge Level
Swaathi, from Garden International School, started by listing the numbers up to 15 and trying to represent them as sums of consecutive numbers:

3 = 1+2
5 = 2+3
6 = 1+2+3
7 = 3+4
9 = 4+5 = 2+3+4
10 = 1+2+3+4
11 = 5+6
12 = 3+4+5
13 = 6+7
14 = 2+3+4+5
15 = 7+8 = 4+5+6 = 1+2+3+4+5

We can't write every number as a sum of consecutive numbers - for example, 2, 4 and 8 can't be written as sums of consecutive numbers. In the above, 9 and 15 were the only numbers that I could find that could be written in more than one way.

Many people spotted the pattern that all odd numbers (except 1) could be written as the sum of two consecutive numbers. For example, Matilda and Tamaris wrote:

If you add two consecutive numbers together, the sum is an odd number, e.g.
and so on...

Well done to pupils from Kenmont Primary School who noticed this, and explained that an Odd plus an Even is always Odd.

Some spotted a similar pattern for multiples of 3. Julia and Lizzie said:

If you add any 3 consecutive numbers together it will always equal a multiple of 3, e.g.

Continuing with the patterns, the Lumen Christi grade 5/6 maths extension program team sent us:

We discovered that the sum of four consecutive numbers gave us the number sequence 10, 14, 18, 22, 26, 30, and so on. They were all even numbers that had an odd number as half of its total.

Heather from Wallington High School for Girls explained this pattern:
10   - 1+2+3+4
14   - 2+3+4+5
18   - 3+4+5+6
22   - 4+5+6+7
In all the columns, each place adds 1 each time, so in total you add 4 each time.

Ruby said:

Numbers which are multiples of 5, starting with 15, are sums of 5 consecutive numbers:

Fergus and Sami noticed a similar pattern:

If you allow negative numbers, you can find a sum for any multiple of 7 easily. Each time you add one number either side of the sum, your sum increases by 7, e.g.

Great! (There's a way to make this pattern work even without using negative numbers - can you spot it?) Why are all these patterns arising?

Becky spotted a different type of pattern:

We found out that powers of 2 (2, 4, 8, 16...) can never be made by adding together consecutive numbers together.

Interesting! I wonder why?

The Lumen Christi team give a way of constructing lots of multiples of odd numbers:

We worked out that if you divide a multiple of 3 by 3, and call the answer n, then your original number is the sum of (n-1), n and (n+1).

Then we discovered that the multiples of 5 can be written as 5 consecutive numbers. It's the same as the rule for 3 consecutive numbers. Take a number and divide it by 5, call it n, and then your number is the sum of (n-2), (n-1), n, (n+1) and (n+2).

We then made a conjecture that since it is true for 3 and 5, it would also work for 7, 9 and any other odd number. We tested it, and it worked. For example, 63 is a multiple of 7 and 9:

7 numbers: 6+7+8+9+10+11+12=63
(63/7 = 9)
9 numbers: 3+4+5+6+7+8+9+10+11=63
(63/9 = 7)

How could we take this investigation further? Arthur asked:

Are there any other patterns?
Can we explore the powers of two further?
Is there a nice way to write certain numbers (for example, every other even number) as a sum of consecutive numbers?

Ottilie suggested:

Instead of adding, you could multiply the consecutive numbers, and see what patterns come up. You could also only add consecutive even numbers, or only consecutive odd numbers. These things could all have something in common, or there could be a pattern between them, or nothing at all, maybe?

Magnus asked:

Is the rule that the powers of two can never be made always true? Can all numbers except the powers of two be made?

Great questions!

By the way, Abhi sent us a nice algebraic proof that powers of 2 can never be made:

Case 1: can we make $2^n$ from an odd number of consecutive numbers?

An odd number of consecutive numbers has a whole number as an average. This average is always the middle number. So, that means that the sum of the numbers will be:

Sum = average $\times$ number of consecutive numbers.
= whole number $\times$ odd number

This means the sum has an odd number as a factor. But $2^n$ cannot have an odd number as a factor. This proves that an odd number of consecutive numbers cannot add to make $2^n$.

Case 2: can we make $2^n$ from an even number of consecutive numbers?

An even number of consecutive numbers will not have a whole number as an average.  The average will be the average of the two middle numbers. So:

Sum = (sum of two middle numbers) $\times \frac{1}{2} \times$ number of consecutive numbers
= (sum of two consecutive numbers) $\times$ ($\frac{1}{2}\times$ Even number)
= (sum of two consecutive numbers) $\times$ whole number

But if you add two consecutive numbers, the answer is always an odd number. So a sum like this must have an odd number as a factor again - but $2^n$ doesn't. This proves that an even number of consecutive numbers cannot add to make $2^n$.

Nicely done!