In my previous post, we found an answer to the question:
What’s the 99th digit to the right of the decimal point in the decimal expansion of ?
However, the solution depended on having the clever idea to add . But there are other ways to come to similar conclusions, and in fact this is not the way I originally solved it.
The first thing I did when attacking the problem was to work out some small powers of by hand:
and so on. It quickly becomes clear (if you have not already seen this kind of thing before) that will always be of the form . Let’s define and to be the coefficients of the th power of , that is, . Now the natural question is to wonder what, if anything, can we say about the coefficients and ? Quite a lot, as it turns out!
We can start by working out what happens when we multiply by another copy of :
But by definition, so this means that and . As for base cases, we also know that , so and . From this point it is easy to quickly make a table of some of the values of and :
Each entry in the column is the sum of the and from the previous row; each is the sum of the previous and twice the previous . You might enjoy playing around with these sequences to see if you notice any patterns. It turns out that there is an equivalent way to define the and separately, such that each only depends on previous values of , and likewise each only depends on previous . I’ll explain how to do that next time, but leave it as a challenge for you in the meantime!