Claire can believe that this will prove that 3 consecutive numbers will add up to a multiple of 3 as the equation 3n+3 (n + n + 1 + n + 2), implies that no matter what number represents 'n' it will be a multiple of three as when you multiply n by 3 you will get a multiple of 3 as your answer, by adding three, it essential is similar to adding 1 to your 'n' (3n+3 = 3(n+1)).
Liz can believe her graph works as the number goes up by the same amount and so if you start with 3 --> 6 --> 9 if you take 3 away from the 9 it makes it 6 and adding the 3 to the 3 which makes all of the numbers now 6 which balances it out.
Charlie's method shows that add all consecutive numbers starting from 1 and regardless of whenever you stop, the answer will always be a multiple of three, this gives him confidence that the consecutive numbers that are brought out randomly will add up to a multiple of 3 due to his theory
When adding 5 consecutive numbers you would get n+n+1+n+2+n+3+n+4 (5n+9) and a irregular pattern that goes from even to odd every time you add 1 to the value of 'n'
When adding 7 consecutive numbers it would be 7n+21 and the number will be a multiple of 7 as 7 is divisible by 7 and 21 is a multiple fo 7 so from that point onwards it is a no brainer