Sums of Consecutive Numbers

Alice has been thinking about sums of consecutive numbers. Here is part of her working out:


9 = 4 + 5 and 2 + 3+ 4
10 = 1 + 2 + 3 + 4
11 = 5 + 6
12 = 3 + 4 + 5
13 = 6 + 7
14 = 2 + 3 + 4 + 5
15 = 7 + 8 and 4 + 5 + 6 and 1 + 2 + 3 + 4 + 5

Rice looked over Alice's shoulder:
"I wonder if we could write every number as the sum of consecutive numbers?"
"Some numbers can be written in more than one way! I wonder which ones?"
"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?"

Let's see if you can answer some of the questions.

Getting Started:
Start by trying some simple cases; perhaps start by exploring what happens when you add two or three consecutive numbers.

1 + 2 + 3 = 6
2 + 3 + 4 = 9
3 + 4 + 5 = 12

Can you explain why the total has gone up by 3 ?
Let us know what you find out. I'll put your explanations with your name on the bottom as answers.