2850.65 – A Sequence by Subtraction

Write down any two numbers aa and bb. Add b−ab - a to the list as the third number, and continue this procedure: each number in the list is the previous number with the number before that subtracted from it. That is how we get b−ab - a for the third number in the list: it's the second number minus the first number. Such a list might go on forever; however, how many different numbers will ever appear in it?

Solution

We try it in general. Here's the list:

abb−a−a=b−a−b−b=−a−(b−a)a−b=−b−(−a)a=a−b−(−b)b=a−(a−b)\begin{array}{cl}a \\b \\b - a \\-a & = b - a - b \\-b & = -a - (b - a) \\a - b & = -b - (-a) \\a & = a - b - (-b) \\b & = a - (a - b)\end{array}

Now we're back at the beginning. That makes six different numbers.