An R.O. for Fractions

I have always believed, contrary to general public opinion, that "FRACTIONS ARE FANTASTIC". I have no great trouble with adding, subtracting, multiplying, or dividing these marvelous little creatures. In fact, I even take a certain amount of pleasure in doing problems with them. Of course, nowadays with the calculators that have the ability to do these computations, well, some of the fear should be removed from those of you out there who don't feel the same as I.

Ah, I guess that's progress; I don't know.

But here is a little Recurrent Operation activity that involves fractions in a most unique and unusual way. Follow the steps that are stated below and you will see for yourself just how fantastic fractions can be at times.

1. Choose any two numbers to start the sequence.

2. Determine every number after the second by increasing the last existing number by one and dividing the sum by the number two places before the term being created.

Here is an example to explain the process. Let's start with 7 and 4. The third term will be 5/7 because you add 1 to the "4" and divide the sum by the "7". The fourth term can now be found by adding 1 to 5/7, then dividing by 4; this yields 3/7.

So far our sequence looks like this:

7,  4,  5/7,  3/7

Now, what would you do next? Of course, add 1 to the fraction 3/7, then divide by the fraction 5/7.

Then continue in like manner until you arrive at a surprising result. (Do you remmember what happened in the activities of Happy & Dizzy Numbers and Ulam?)

Of course, you're curious if this will always happen every time, right? Well, try some more numbers. Try beginning with fractions for your first two selections; even mix in some negative numbers, if you feel adventuresome. That's the way you really learn something in math: try it and see.

One warning here: just looking at many, many examples is not a mathematical proof that the surprising event that you have noticed by now always will occur. That, my friend, is where algebra comes to our aid! Find an algebraic proof that this activity produces a certain interesting outcome and email it to me. (See link below.)

By the way, can you find two starting numbers, whole numbers that is, that never bring about fractions as you build the sequence? It can be done.