Last week we looked at examples of induction proofs: some sums of series and a couple divisibility proofs. This time, I want to do a couple inequality proofs, and a couple more series, in part to show more of the variety of ways the details of an inductive proof can be handled.
Last week we looked at introductory explanations of what mathematical induction is, including answers to some misunderstandings of the concept. But we only looked at one trivial example of such a proof; for a real understanding of the technique, we need some fuller examples. For that purpose, I have chosen a few questions we have …
We’ve been looking at dissection puzzles, where we cut an object into pieces, and rearrange them. Here we’ll examine a mystery posed by two different puzzles, each of which seems to change the area by rearranging the pieces. The answer combines the marvelous Fibonacci numbers and [spoiler alert!] how easily we misjudge areas.
(A new question of the week) We usually see limits applied to functions in a calculus class. An interesting question from late October deals with a limit in a geometrical construction based on a function. We’ll be seeing how to discover a proof, then several alternative proofs, and finally what the answer means.
(A new question of the week) A question from last month provides an opportunity to show how to develop an algebraic proof of a combinatorial identity involving factorials. We’ll be looking over Doctor Rick’s shoulder as he guides a student through the maze. I’ll also add in a previously published version of the same proof …
(A new question of the week) I have often said that calculus class is where many students finally learn algebra, because now algebra is an essential tool, not just something to learn for an exam. This is especially true of a nontraditional student, who may not have taken math recently, or may even be learning …
One thing we enjoy doing is guiding a student through the process of problem-solving. Here is a problem from August that illustrates how to think through a complicated geometrical proof. In particular, this uses some circle theorems involving chords, secants, and tangents, together with a bit of algebra.
(A new question of the week) A question from the end of August led a student and a Math Doctor to an extra challenge, by way of an apparent typo in the problem. We particularly enjoy working with students who are willing to take on extra work in order to learn more than they need …
(A new question of the week) Some questions we get, while small, raise interesting issues. In a question we got last month, there are several little issues pertaining to how the final answer should be chosen; as is too often the case, it seems that a diligent student who cares about accuracy might be penalized. …
A Test Dilemma: Do As You’re Told, or Do What’s Right? Read More »
Last week we looked at counting the maximum number of pieces into which a circle can be cut by n chords (straight lines). Here we will look at a similar-sounding problem where we use all the chords formed by n points on the circle. We’ll also see an important example of why we shouldn’t jump …