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Activity for Carefree Explorer‭

Type On... Excerpt Status Date
Edit Post #287063 Initial revision over 1 year ago
Question Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$.
> Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$. This was a simple observation I made when playing around and I came up with the following proof: > It follows from $(10a + 25)^2 = 100a^2 + 500a + 625 = 100(a^2 + 5a) + 625$ that any number ending in $25$ raised to a p...
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over 1 year ago
Comment Post #286908 Some insightful comments; thank you. You may want to consider moving some of it into your answer.
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over 1 year ago
Comment Post #286908 Thank you for the comments on proof writing; I will certainly keep them in mind. For inducting over the reals, I tried to show that $P(0)$ holds and that $\forall x \\; P(x) \implies P(x \pm \lim_{\epsilon \to 0}\epsilon)$ where $P(x): e^x \ge x + 1$ and $\lim_{\epsilon \to 0}\epsilon$ represents som...
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over 1 year ago
Edit Post #286907 Post edited:
over 1 year ago
Edit Post #286907 Initial revision over 1 year ago
Question Prove $e^x \ge x+1 \\\; \forall x \in \mathbb{R}$ using induction
> (How) can we prove $e^x \ge x+1 \\; \forall x \in \mathbb{R}$ using induction (without using the derivative of $e^x$ at any stage)? Comments on my attempt are appreciated. I stumbled across a very nice proof of $\frac{\mathrm{d}}{\mathrm{d}x}e^x = e^x$ that uses the identity $e^x \ge x+1$. Br...
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over 1 year ago
Comment Post #286848 Thank you for the feedback. I have edited my post with clarifications.
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over 1 year ago
Edit Post #286848 Post edited:
over 1 year ago
Edit Post #286848 Initial revision over 1 year ago
Question Is there a way to encode a unique arrangement of vertices of a graph with a unique short word?
I call graphs $G1$ and $G2$ distinct iff (i) $G1$ has a different arrangement1 of vertices than $G2$ and (ii) $G1$ and $G2$ have the same number of vertices. All other properties of $G1$ and $G2$ have no effect on distinguishing them. For example, $G1$ and $G2$ may be the same even if the number of e...
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over 1 year ago
Comment Post #286832 Thank you, that’s insightful.
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over 1 year ago
Comment Post #286833 I wish I could upvote this a 100 times :)
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over 1 year ago
Edit Post #286829 Initial revision over 1 year ago
Question Show that $f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2}$
> Show that $$f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2} \quad \forall x \in (-\infty, -1)\large\cup (0, \infty)$$ I observed this feature graphically, but am looking for a way to prove it. My attempt is as follows: > We can show that $$\arc...
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over 1 year ago