What does it all add up to?
$5, 6, 7, 8$ are four consecutive numbers. They add up to $26$.
Take other sets of four consecutive numbers and find their total.
Do the totals have anything in common?
Can you find four consecutive numbers that add to $80$?
If not, might it be impossible?
What other even numbers cannot be written as the sum of four consecutive numbers?
Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true, and look to proofs to provide rigorous and convincing arguments and justifications.
Can you prove that the sum of four consecutive numbers is always an even number which is not a multiple of $4$?
Below is a proof that has been scrambled up.
Can you rearrange it into its original order?
Click on student solutions to see some different proofs that students submitted.
Extension:
Can you prove that the sum of five consecutive numbers is always a multiple of $5$?
Can you prove that the sum of six consecutive numbers is always a multiple of $3$ which is not a multiple of $6$ (i.e. an odd multiple of $3$)?
Challenging Extension:
Can you prove the following statements?
- If $n$ is odd, then the sum of $n$ consecutive numbers is always a multiple of $n$.
- If $n$ is even, then the sum of $n$ consecutive numbers is always a multiple of $\dfrac{n} {2}$, but is not a multiple of $n$.
Tobias from Wilson's School sent in this diagrammatic proof that the sum of four consecutive numbers can never be a multiple of 4:
Shayen from Wilson's School in England investigated the sums of 5 consecutive numbers. Here is Shayen's work (click to see a larger version):
Tobias sent in this diagrammatic proof that what Shayen noticed is always true:
Sunhari from British School Muscat and Mahdi from Mahatma Gandhi International School in India sent in this algebraic proof:
Let the five consecutive numbers be $n, n+1, n+2, n+3, n+4$
$\begin{split}n+n+1+n+2+n+3+n+4 & =5n+10\\
&=5(n+2)\\
&=5a for a = n+2\end{split}$
Shayen also investigated the sums of 6 consecutive numbers. Here is Shayen's work (click to see a larger version):
Here is Tobias's diagrammatic proof of what Shayen noticed:
Here is Mahdi and Sunhari's algebraic proof:
Let the six consecutive numbers be $n, n+1, n+2, n+3, n+4, n+5$
$\begin{split}n+n+1+n+2+n+3+n+4+n+5&= 6n + 15\\&= 3(2n+5)\end{split}$
It is an odd multiple of 3, and cannot be factorised to a multiple of six.
Shayen made a conjecture about adding other numbers of consecutive numbers (click to see a larger version):
Egeman from Wilson's School represented this algebraically:
Let $a$ be the first number in a sequence of consecutive numbers
Let $n$ be the number of numbers in a sequence of consecutive numbers
Let $s$ be the sum of the numbers in a sequence of consecutive numbers
where $n=1, s=1a+0$;
where $n=2, s=2a+1$;
where $n=3, s=3a+3$;
where $n=4, s=4a+6$;
where $n=5, s=5a+10$;
where $n=6, s=6a+15$; etc
$s$ consists of two parts: a multiple of $a$ and a triangle number.
The multiple of $a$ is always $na$.
The triangle number is the $(n-1)^\text{th}$ term in the sequence of triangle numbers.
Mahdi and Sunhari showed why $s$ takes this form. This is Mahdi's work:
From here, Mahdi, Sunhari and Egeman all used a formula for the $(n-1)^\text{th}$ triangular number to show that if $n$ is odd, $s$ is a multiple of $n$, and if $n$ is even, then $s$ is not a multiple of $n$ but is a multiple of $\frac n2.$ Here is Mahdi's work:
Why do this problem?
This problem, along with the rest of the problems in the Proof for All (st)ages feature, provides an excellent context for observing, conjecturing and thinking about proof, and for appreciating the power of algebra.
Possible approach
This problem featured in an NRICH Secondary webinar in January 2022.
These printable cards for sorting may be useful:
What Does it All Add Up To Proof Sort
Key question
Is there a way to represent the sum of four consecutive numbers that will help to explain the patterns you noticed?
Possible support
Encourage students to work in pairs on the proof sorting exercise.