Challenge Level

5, 6 and 7 are three consecutive numbers. They add up to 18.

14, 15 and 16 are also three consecutive numbers. They add up to 45.

**Take other sets of three consecutive numbers and find their total.**

What do you notice?

Do the totals have anything in common?

How can you be sure that what you have noticed will always be true?

Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true.

Have you been able to provide an argument that would convince mathematicians?

Liz noticed that all the totals are a multiple of 3. She found it useful to draw a picture:

**How did this help to convince Liz that three consecutive numbers always add up to a multiple of 3?**

Charlie also noticed that all the totals were a multiple of 3. He thought about sets of numbers in a systematic way:

Charlie started with the three consecutive numbers 1, 2, 3. They add up to 6.

He added 1 to each, which gave him the next three consecutive numbers 2, 3, 4. They add up to 9.

He added 1 to each again, and ended up with 3, 4, 5. They add up to 12.

**How did this help to convince Charlie that three consecutive numbers always add up to a multiple of 3?**

If you have met algebra before, then you might like to look at Claire's method:

Claire decided to call the first number $n$.

The next two numbers are then equal to $n+1$ and $n+2$.

Claire added the three numbers to get $n+ n+1 + n+2 = 3n + 3$.

**What happens when you add five consecutive numbers? Seven consecutive numbers? ...**