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 $n$.