1:"I wonder if we could write every number as the sum of consecutive numbers?’’
The answer to this question is NO. There are some obvious numbers that cannot be made by using consecutive numbers, like 0, but there are some others, like 2, and 4. It didnt take very long to work this out as it was most logical to look at the smallest numbers that couldn’t be made. It is possible to write every odd number as a sum of consecutive numbers, but as for the even numbers, it must be divisible by and odd number, because you nedd two odd numbers to make an even one.
2: "9 ,12 and 15 can all be written using three consecutive numbers. I wonder if all multiples of 3 can be written in this way?"
YES. All multiples of 3 can be made with 3 consecutive numbers. A clear indication of this is shown with these multiples of three:
3 = 0+1+2 12=3+4+5 21=6+7+8
6=1+2+3 15=4+5+6 24=7+8+9
9=2+3+4 18=5+6+7 27=8+9+10
This can be expressed in algebra like this:
n=3x
n=(x-1)+x+(x+1)
3: "Maybe you could write the multiples of 4 if you used four consecutive numbers..."
You CANNOT write the multiples of 4 with four consecutive numbers as the numbers 4 and 8 cannot be made with four consecutive number, let alone two or three consecutive numbers, because 4 and 8 are even numbers that cannot be divided by and odd number.