You all probably know what an arithmetic sequence is. It's a sequence of numbers such that consecutive terms always have the same difference. But, there is another, more general definition: An arithmetic sequence is a sequence of real numbers such that every term in the sequence is the arithmetic mean of the preceding and the following term.
But what if we take some other mean instead of the arithmetic mean?
Other well known means are the geometric mean, the harmonic mean and the quadratic mean.
The geometric mean of two numbers and is . Applied to our definition for a (now geometric) sequence , this means
If we define , we see , which is another known definition of a geometric series.
The harmonic mean of two numbers and is defined as . To get a recursive formula,
But this is not what I am aiming for. These means are too restrictive. However, there js a generization that involves all of these means as special cases. This is called the Hölder mean and it generalizes by introducing a new parameter . In the case of 2 numbers and , their Hölder mean with parameter is defined as
With these means, we can also generalize our sequences to general "p-mean sequences".
In such a sequence, the equation has to hold for all .
We can rearrange this equation to get a recursive equation for any p-mean sequence
To get a feeling of these sequences, let's look at some examples. For simplicity, we will always take and .
Since these numbers aren't that "nice" (because they involve roots), let's define another sequence to get rid of these roots.
We can adopt the definition of to get a recursive equation for
The starting values are and .
Let's look at the table again, this time also with .
We observe something interesting; is always an arithmetic sequence. To get its general formula, let's find the formula for each individually
It looks like the general equation is .
Let's prove this by induction
Base case 1:
Base case 2:
Suppose holds true for some . Then, by the induction hypothesis (i.h.), it must also be true for .
Since we end up with in the end and used the formula for and the definition of a p-mean sequence, this completes our proof for the formula of such a p-mean sequence