Summing consecutive numbers
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Problem
Summing Consecutive Numbers printable worksheet
Watch the video below to see how numbers can be expressed as sums of consecutive numbers.
Investigate the questions posed in the video and any other questions you come up with.
Can you draw any conclusions?
Can you support your conclusions with convincing arguments or proofs?
If you are unable to view the video, click below to reveal an alternative version of the problem.
Charlie has been thinking about sums of consecutive numbers. Here is part of his working out:
Alison looked over Charlie's shoulder:
"I wonder if we could write every number as the sum of consecutive numbers?"
"Some numbers can be written in more than one way! I wonder which ones?"
"$9$, $12$ and $15$ can all be written using three consecutive numbers. I wonder if all multiples of $3$ can be written in this way?"
"Maybe you could write the multiples of $4$ if you used four consecutive numbers..."
Choose some of the questions above, or pose some questions of your own, and try to answer them.
Can you support your conclusions with convincing arguments or proofs?
Getting Started
Start by trying some simple cases; perhaps start by exploring what happens when you add two or three consecutive numbers.
$1 + 2 + 3 = 6$
$2 + 3 + 4 = 9$
Can you explain why the total has gone up by $3$?
Student Solutions
Swaathi, from Garden International School, started by listing the numbers up to 15 and trying to represent them as sums of consecutive numbers:
2
3 = 1+2
4
5 = 2+3
6 = 1+2+3
7 = 3+4
8
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.
1+2=3
2+3=5
3+4=7
4+5=9
5+6=11
6+7=13
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.
1+2+3=6
2+3+4=9
3+4+5=12
4+5+6=15
5+6+7=18
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.
1+2+3+4=10
2+3+4+5=14
3+4+5+6=18...
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:
1+2+3+4+5=15
2+3+4+5+6=20
3+4+5+6+7=25...
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.
3+4=7
2+3+4+5=14
1+2+3+4+5+6=21
0+1+2+3+4+5+6+7=28
-1+0+1+2+3+4+5+6+7+8=35...
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!
Teachers' Resources
Why do this problem?
This problem offers a simple context to begin exploration that naturally leads to some interesting conjectures about properties of numbers, and the possibility of developing some quite sophisticated algebraic arguments and proofs.
Possible approach
This printable worksheet may be useful: Summing Consecutive Numbers.
One approach is to introduce the lesson in the way it is introduced on the video, with students suggesting numbers and then the teacher at first, and then the whole class, finding ways to express them as consecutive sums.
Here is an alternative approach:
"Can someone give me a set of two or more consecutive numbers?"
Write a few sets on the board.
"What totals do we get by adding the consecutive numbers in these sets?"
Write + signs in between the lists of numbers.
"These totals are all examples of numbers that can be written as the sum of consecutive numbers. I wonder if all numbers can be written in this way ”¦"
"How about trying to write the numbers from $1$ to $30$ as sums of consecutive numbers?"
Give students time to work in pairs on filling in the gaps from $1$ to $30$. While they are working, write the numbers from $1$ to $30$ on the board ready to collect together the sums the class have found.
If students ask about negative numbers, one possible answer is: "Stick to positive numbers for now, and then perhaps investigate negative numbers later."
Once most pairs have filled in most of the gaps, collect their results on the board.
"Spend a minute looking at these results and then be prepared to talk about anything interesting that you notice."
Give them time to think on their own at first and then share ideas with their partner, before discussion with the whole class.
Next, collect together any noticings, and write them on the board in the form of questions or conjectures. If such conjectures are not forthcoming, here are some suggested lines of enquiry:
• "I wonder if we could write every number as the sum of consecutive numbers?"
• "Some numbers can be written in more than one way! I wonder which ones?"
• "9, 12 and 15 can all be written using three consecutive numbers. I wonder if all multiples of 3 can be written in this way?"
• "Maybe you could write the multiples of 4 if you used four consecutive numbers ..."
Allow pairs time to work on the conjectures of their choosing, reminding them that they will need to provide convincing arguments to explain any of their conclusions.
If appropriate, bring the class together to spend some time discussing algebraic representations of consecutive numbers ($n, n+1, n+2...$) to give students the tools to create algebraic proofs.
Finish the lesson by challenging students to express numbers that you have chosen as a sum of consecutive numbers in as many ways as possible - e.g. 45 can be expressed as the sum of consecutive numbers in 5 different ways.
Key questions
Possible support
| 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 |
Possible extension
Challenge students to find an efficient way to calculate how many different ways $x$ can be expressed as a sum of consecutive numbers, for any $x$.