# Tag Info

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### Does a proof by induction have to explicitly refer to the principle of mathematical induction?

The appropriate level of granularity for a proof depends on the audience. If you're taking an "Intro to Proofs" class and your homework is to do some proofs by induction, then yeah, you ...
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Accepted

### Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour

I looked at some of your posts on MSE before answering. Well, I wouldn't say that you are a "Jack of trades" yet, but you are certainly way above what one would expect from somebody 2 years ...
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### Best practices for Proof Revision/ Proof Portfolio?

I taught an abstract algebra course this way. Each student made an overleaf account which I required to be something like "Firstname-Lastname-MTH358-Spring-2024". They shared this folder ...
• 25.5k
Accepted

### How to prove, without the LOTUS formula, to student that $V[aX+b]= a^2 V[X]$?

I think you will want to start by convincing the audience that $p(aX+b = ax_i + b)$ is equal to $p(X=x_i)$, probably with examples. I am not an expert statistician so please let me know politely if I ...
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### How to convince a student without calculus that great circles are geodesics in a sphere?

Take a physical sphere such as a beach ball, and a string. Pick two points. Hold the string down with one finger at one point then stretch it to the second point. Next, holding the string tight at ...
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### Does a proof by induction have to explicitly refer to the principle of mathematical induction?

Typically you want to name things. This makes them visible and something you can discuss. So, while teaching, you do want to say that this thing her is induction, so we have to remember to check the ...
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### Identifying Trigonometrical proofs

The proof: https://youtu.be/p6j2nZKwf20 For context, here's the main idea of the proof. Using the definition of sine, we have $c^2 = \dfrac{2ab}{\sin 2\alpha}.$ Our goal is to show that the RHS is ...
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### Does a proof by induction have to explicitly refer to the principle of mathematical induction?

Peano's axioms without the axiom of induction has some models that do not correspond to the natural numbers that we have in our minds. In order to prove that every natural number $n$ other than $0$ ...
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### Does a proof by induction have to explicitly refer to the principle of mathematical induction?

In a pedagogical context I can see four types of situation where it may reasonably be required for students to explicitly state their use of the principle of mathematical induction: If writing proofs ...
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### Teaching strong induction instead of induction

Note that the "strong induction" is not even the most general induction scheme you can design. A high school level example is the proof of the AM-GM inequality for $n$ positive numbers where ...
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### How to convince a student without calculus that great circles are geodesics in a sphere?

Determine the shortest route from New York City USA to Rome Italy using a piece of string on a globe. Both cities are south of the 45th parallel and yet the the shortest route deviates considerably to ...
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Let's not overlook an obvious application of induction is to turn it around and go from general to specific. Let's say we have established $H(1)$ and that $H(k+1)$ follows from $H(k)$ and we are ...
To be a proof, an argument needs to be explicit about the logical structure. An induction proof won't be any different. Take a very standard task like: such as showing that, for all $n\in \mathbb{N}$...