A bit about me
Go Bruins (and Grizzlies)!
“Making mistakes is what is expected in order to learn, being wrong about math isn't wrong, but actually right only if we correct it.” Dr. David Erickson
Forget the mistaken idea that an answer in math is simply right or wrong. We are to learn to think mathematically and concentrate on the process.
“It is the story that matters not just the ending.” Paul Lockhart.
Monday, September 2, 2019
Let a_(n+1)=sqrt(2+a_n) with a_1=sqrt(2). Find a_n as n goes to infinty.
Using some nimble algebraic manipulations, the Calculus class showed the limit to be 2. The Precalc class was able to arrive at the same conclusion through investigating the sequence using a spreadsheet.
The next obvious question was:
What are the requirements on positive integer values of p so that the limit converges to a whole number? That is, when will a_(n+1)=sqrt(p+a_n) with a_1=sqrt(p) converge to a whole number?
The precalc students showed that this happens precisely when p is of the form m(m-1) for positive integer values of m. Further, if p is of the form m(m-1), then a_n will equal converge to m as n grows large.
This intersects nicely with the work of the calc students who were able to show that a_(n+1)=sqrt(p+a_n) with a_1=sqrt(p) converges to [1+sqrt(1+4p)]/2. Outstanding work, though they were left wondering the requirements on p for (1+4p) to be a perfect square. The precalc class was able to supply this answer. If the sequence converges to a whole number m for p of the form m(m-1), then [1+4m(m-1)] must be a perfect square!
A simple investigation into infinite nested radicals led to some number theory discoveries. As usual when attempting to answer a question more questions have to be asked.
(1+4p) being a square root, doesn't ensure the sequence converges to a whole number. Or does it?
Certainly you can find a whole number p to so that the sequence will converge to any whole number m. Or can you?
Does their exist a whole number p so that the sequence will converge to 1?
Lots of loose ends here, I fear they may have lost interest...
Tuesday, August 20, 2019
Mistakes are a necessary component of learning. The history of mathematics as a discipline is no different. It is rife with mistakes, wrong conclusions, guesses, controversies, conspiracies, secrets, and murder. It is a messy, uncertain business. We will allow it to be so as well; without, of course, the conspiracy, secrecy, and murder! Often, we will have an end in sight, but the journey is perhaps more important than the destination. The great mathematician Karl Friedrich Gauss said, “I have had my results for a long time: but I do not yet know how I am to arrive at them.” Many times in life, we have a desired solution or outcome but are at pains with just how to realize it. GK Chesterton says of people in such a place, “It isn’t that they cannot see the solution. It is that they cannot see the problem.” If this is true, then strategies, methods, and ways of thinking that allow us to make sense of the problem while giving direction towards resolution are the keys to mathematical proficiency. This is what we are about here. “It is the story that matters not just the ending.” Paul Lockhart.
Wednesday, August 14, 2019
Pafnuty Chebyshev 1821-1894
Tuesday, December 12, 2017
Monday, December 4, 2017
Wednesday, February 15, 2017
Saturday, February 11, 2017
If n is the nth term in the Fibonacci Sequence and n+1 is the nth plus one term, Then as n goes to infinity, the ratio of n+1/n converges to the Golden Ratio. We also found that this is the case with the Lucas Sequence.
Under current investigation is the question of what types of numbers must be our first and second term in the sequence for this surprising property to hold. That is, does always hold when the first terms are positive integers? integers in general? rationals? irrationals? transcendentals?
Ask a programmer about these questions, they are exploring some good stuff!