Just an "Average" Proof

I honestly have no idea how this problem came to mind. Oh well; here it goes.

Start with a sequence \(a_n=\frac{a_{n-1}+a_{n-2}}{2}\), where the each term is the average of the previous terms. Define a function \(f\) that chooses the base terms, \(a_1\) and \(a_2\),: \[f(x,~y)=\begin{cases} x=a_1 \\ y=a_2 \\ a_n=\frac{a_{n-1}+a_{n-2}}{2} \\ \displaystyle\lim_{n\rightarrow \infty}a_n \end{cases}\]

After coding up the function in Python and testing a few cases, I conjectured \((i.)\):

If \(f(x,~y)=\begin{cases} x=a_1 \\ y=a_2 \\ a_n=\frac{a_{n-1}+a_{n-2}}{2} \\ \displaystyle\lim_{n\rightarrow \infty}a_n \end{cases}\), then \(f(x,~y)=\frac{x+2y}{3}\).

It is interesting to note that the proposed function is the average of the numbers \(x,~y,\) and \(y\). 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: \[x,~y,~\frac{x+y}{2},~\frac{\frac{x+y}{2}+y}{2},~\frac{\frac{x+y}{2}+x+2y}{4},~\frac{\frac{x+y}{2}+2x+5y}{8},\dots\] A pattern is most definitely emerging:

  1. The third term \(\left(\frac{x+y}{2}\right)\) is always present in the sequence.

  2. The denominator seems to be increasing in powers of 2 (the algebra shows why).

  3. The \(y\)-coefficient in the \(n\)th term is the \(x\)-coefficient in the \(n-1\)th term (looking at the Python program shows why this is true).

Next, find the recursive formula for the \(y\)-coefficient of the \(n\)th term. Work: \[\dfrac{\dfrac{\frac{x+y}{2}+x+2y}{4}+\dfrac{\frac{x+y}{2}+2x+5y}{8}}{2}=\dfrac{\dfrac{x+y+2x+4y+\frac{x+y}{2}+2x+5y}{8}}{2}=\dfrac{\frac{x+y}{2}+5x+10y}{16}\]

Firstly, multiply first term by \(\frac{2}{2}\), and then combine like terms. The new \(y\)-coefficient is the sum of the previous coefficient, double the coefficient before that, and 1. In other words, \(c_n=c_{n-1}+2c_{n-2}+1\), with \(c_1=c_2=0\).

Since the denominator of the \(n\)th term is \(2^{n-3}\), that means \[(ii.)~f(x,~y)=\frac{x+2y}{3}\iff \displaystyle\lim_{n\rightarrow \infty}\frac{c_n}{2^{n-3}}=\frac{2}{3}\] In other words, in order for the function to be true, the \(y\)-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 \(\displaystyle\lim_{n\rightarrow \infty} \frac{c_n}{2^{n-3}}=L\). Secondly, it must be acknowledged that the \(x\)-coefficient is, as established earlier, \(c_{n-1}\). Therefore, the \(x\)-coefficient limit is \[\displaystyle\lim_{n\rightarrow \infty} \frac{c_{n-1}}{2^{n-3}}=\displaystyle\lim_{n\rightarrow \infty} \frac{c_n}{2^{n-2}}=\displaystyle\lim_{n\rightarrow \infty} \frac{c_n}{2^{n-3}\cdot 2}=\frac{L}{2}\]

Next, find the sum of these two limits; the \(y\)-limit is the same as ignoring the \(x\) value of the function and plugging in 1 for \(y\) (since 1 is the multiplicative identity); the opposite is true for the \(x\)-limit. Therefore, the sum of these to limits is the output of the function when \(x=y=1\):

\[\implies L+\frac{L}{2}=f(1,~1)\]

Recall that \(f\) was the limit of a recurring sequence that averaged 2 terms. Consider \(f(b,~b)\): taking the average of 2 identical numbers outputs that number. If the process is repeated infinitely many times, it will make no difference. Therefore, \[L+\frac{L}{2}=f(1,~1)=1\] \[\frac{3}{2}L=1\] \[L=\frac{2}{3}\] Therefore, by \((ii.)\), the original conjecture is true!

\[\huge{\beta_{\lceil \mid \rceil}}\]

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. 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.

Note by Blan Morrison
5 days, 4 hours ago

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Interesting! If you add in a little linear algebra, you would be able to generalise to more linear recurrence relations of higher order, and even know whether it converges at all:

You can split your recurrence relation into multiple degree 1 recurrence relations. For your recurrence this goes from \(\displaystyle a_n=\frac{a_{n-1}+a_{n-2}}{2}\) to:

\[\begin{align*} b_n&=a_{n-1} \\ a_n&=\frac{a_{n-1}+b_{n-1}}{2} \end{align*}\]

The reason for doing this is so that you can express the nth term of the recurrence relation as a matrix product:

\[\begin{align*} \begin{pmatrix} a_0 \\ b_0 \end{pmatrix} &= \begin{pmatrix} y \\ x \end{pmatrix} \\ \begin{pmatrix} a_n \\ b_n\end{pmatrix} &= \begin{pmatrix} 0.5 & 0.5 \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} y \\ x \end{pmatrix} \end{align*}\]

Let's make \(\displaystyle \begin{pmatrix} 0.5 & 0.5 \\ 1 & 0 \end{pmatrix} = A\) and \(\displaystyle \begin{pmatrix} y \\ x \end{pmatrix} = \vec x\) for notation sake. If we want to find \(a_n\) as \(n \rightarrow \infty\), then we want to find \( \displaystyle \lim_{n \rightarrow \infty} A^n \vec x\)

We can diagonalise the matrix \(A\) using its eigenvalues and eigenvectors. For this question,

\[A = PQP^{-1} = \begin{pmatrix} 1 & -0.5 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -0.5 \end{pmatrix} \begin{pmatrix} 1 & -0.5 \\ 1 & 1 \end{pmatrix}^{-1} \]

And the reason why we diagonalise it is because we can easily calculate \(A^n\):

\[A^n = PQ^nP^{-1} = \begin{pmatrix} 1 & -0.5 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1^n & 0 \\ 0 & (-0.5)^n \end{pmatrix} \begin{pmatrix} 1 & -0.5 \\ 1 & 1 \end{pmatrix}^{-1} \]

It is here we can see if \(a_n\) diverges. If it does, the terms in \(Q^n\) would blow up, making \(A^n \vec x\) blow up as well. Luckily, (and expectedly), for this value of \(A\), the terms in Q converge, making:

\[\lim_{n\rightarrow \infty} A^n = \begin{pmatrix} 1 & -0.5 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & -0.5 \\ 1 & 1 \end{pmatrix}^{-1}\]

And so the coveted value we wanna find can be calculated by:

\[\begin{align*}\begin{pmatrix} a_n \\ b_n\end{pmatrix} &= \begin{pmatrix} 1 & 1 \\ -0.5 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -0.5 & 1 \end{pmatrix}^{-1} \begin{pmatrix} y \\ x \end{pmatrix} \\ &= \frac{1}{3} \begin{pmatrix} 2x+y \\ 2x+y \end{pmatrix} \end{align*}\]

Interestingly, in general, \(a_n\) is likely to diverge or go to 0. This is because in order for \(a_n\) to converge to a non-zero value, \(Q\) must have an entry of value \(1\), or \(Q^n\) diverges or becomes the null matrix. You are very lucky to have chanced upon this recurrence relation.

Oh and as for the \(\LaTeX\), you can use \begin{align} <ur eqns> \end{align} to line them up neatly.

Julian Poon - 1 day, 20 hours ago

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To add to the last point: You'd want the eigenvalues of \(Q\) to all be between \(-1\) and \(1\) (inclusive, I think?), otherwise the limit of the SVD approaches infinity.

On a more sour note: Can you check your calculations again? The matrix \(A\) should read

\( \begin{pmatrix} 0 & 1 \\ 0.5 & 0.5 \end{pmatrix} \)

if you define the recurrence as such. If we modify Blan's initial terms to \(a_0\) and \(a_1\) instead of \(a_1\) and \(a_2\) then you would want the matrix equation to start at \(n = 1\), so that your initial terms are given by \(b_1\) and \(a_1\), not \(b_0\) and \(a_0\). Aside from that, maybe you could check your SVD again, but it shouldn't change the \(Q\) matrix if you modify the \(P\) matrix. I'd also put a \(\lim\) sign in front of the \(b_n\), \(a_n\) vector at the end, to be precise.

Gennady Notowidigdo - 4 hours ago

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My bad, this can be easily remedied by switching the positions of \(a_n\) and \(b_n\), a typo on my part. If we want \(a_n\) to converge to a finite non zero value, all the entries of \(Q\) have to be between -1 and 1, inclusive of \(1\) but not inclusive of \(-1\). Furthermore, at least 1 entry of \(Q\) has to be \(1\). With regards to the undex starting from \(0\), it is essentially the same, but i chose to start from 0 so that I don't have a \(n+1\) power in my \(Q\), which is kinda ugly

Julian Poon - an hour ago

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@Julian Poon No worries. Just wanted to make sure your notation is consistent with Blan’s.

Gennady Notowidigdo - 43 minutes ago

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I'm confused when you say "generalize." Do you mean the average of the previous \(n\) terms instead of 2? It looked to me as if you did something different, but I honestly have little to no comprehension of linear algebra (yet).

Blan Morrison - 23 hours ago

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Have a look into linear recurrence relations and how to solve them in general.

Gennady Notowidigdo - 5 hours ago

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By generalise I mean any recurrence relation of the form \( \displaystyle a_n = \sum_{i=1}^{k} c_i a_{n-i} \)

Julian Poon - 22 hours ago

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@Julian Poon Ah, okay.

Blan Morrison - 14 hours ago

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A really interesting note! Have you thought about generalising what happens with this sequence for an \(n^{\text{th}}\)-order linear recurrence relations? Would be interesting to see what happens...

Gennady Notowidigdo - 4 days, 20 hours ago

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I was just thinking about that this morning! I'll definitely make an update about that sometime in the next 12 hours.

Blan Morrison - 4 days, 14 hours ago

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