I honestly have no idea how this problem came to mind. Oh well; here it goes.
Start with a sequence , where the each term is the average of the previous terms. Define a function that chooses the base terms, and ,:
After coding up the function in Python and testing a few cases, I conjectured :
If , then .
It is interesting to note that the proposed function is the average of the numbers and . While it seems pretty obvious that this is true, nothing is true in mathematics until it is proven. A proof by induction probably seems like the most logical method, but I don't really know how to perform such a proof, much less with 2 variables. So, I tried deriving the formula.
Start with the first few terms of the sequence: A pattern is most definitely emerging:
The third term is always present in the sequence.
The denominator seems to be increasing in powers of 2 (the algebra shows why).
The -coefficient in the th term is the -coefficient in the th term (looking at the Python program shows why this is true).
Next, find the recursive formula for the -coefficient of the th term. Work:
Firstly, multiply first term by , and then combine like terms. The new -coefficient is the sum of the previous coefficient, double the coefficient before that, and 1. In other words, , with .
Since the denominator of the th term is , that means In other words, in order for the function to be true, the -coefficient in the algebraic expansion needs to be two-thirds of the denominator as the order of terms gets larger. (This is the link that can prove the function!)
Firstly, define . Secondly, it must be acknowledged that the -coefficient is, as established earlier, . Therefore, the -coefficient limit is
Next, find the sum of these two limits; the -limit is the same as ignoring the value of the function and plugging in 1 for (since 1 is the multiplicative identity); the opposite is true for the -limit. Therefore, the sum of these to limits is the output of the function when :
Recall that was the limit of a recurring sequence that averaged 2 terms. Consider : taking the average of 2 identical numbers outputs that number. If the process is repeated infinitely many times, it will make no difference. Therefore, Therefore, by , the original conjecture is true!
Please feel free to critique this proof; I am here to learn! I feel that it is too unprofessional, but I am still proud of it; after writing the proof, I googled "recursive sequence limits" and found a much shorter proof on StackExchange, so I know this isn't the best possible proof. I feel like such an amateur after writing this because it is so informal, and I wonder if this what being a "real" mathematician is like. Of course, it probably isn't and I'm just being a kid with excessively high hopes.