The answer to Alison’s question: ‘I wonder if we could write every number as a sum of consecutive numbers’ is no and this is proved very simply by taking the number 4. 4 cannot be written in consecutive numbers because 1+2= 3 (not high enough), 1+2+3= 6 (too high), 2+3=5 (too high).
Because the random number, 4, cannot be written as a sum of consecutive numbers, this proves that not EVERY number can be written as a sum of consecutive numbers.
To the question: ‘I wonder if all multiples of 3 can be written using 3 consecutive numbers?’, the multiples of 3 follow a pattern.:
The first multiple, 3, can be written as 0+1+2
The next multiple, 6, can be written as 1+2+3, which uses the MIDDLE consecutive number of 3, 1, as its FIRST consecutive number.
The following multiple of 3, 9, can be written as 2+3+4, again using 2, the MIDDLE consecutive number of 6.
Therefore, if this pattern is followed for each multiple of three, only three consecutive numbers need to be used – the first being the MIDDLE consecutive number of the previous multiple of three.
Another question which arises over the matter of consecutive numbers may be:
“Can any number, written in an amount of consecutive numbers equal to its value, result in a multiple of that same number?” For example, when 5 is written as 1+2+3+4+5, will the answer be a multiple of 5? Yes it can, because 1+2+3+4+5= 15, which is a multiple of 5.
Let’s try this theory with another number. Take 4, for example. 1+2+3+4= 10. 10 is NOT a multiple of 4, therefore this theory does not work for any number.