Activity for Carefree Explorer‭

Type	On...	Excerpt	Status	Date
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... (more)	—	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... (more)	—	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... (more)	—	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... (more)	—	over 1 year ago