Articles of algebra precalculus

Show that ${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$

This question already has an answer here: Evaluating even binomial coefficients 5 answers Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$ 7 answers

Solve discrete Math Problem using abstract algebra, postage problem?

The question I am looking at is not very hard: Determine which amounts of postage can be written with $5$ and $6$ cent stamps. To determine the amount, use a brute force way to solve it. Counting from $0$, see if each number can be written with $5$ and $6$. I can get $20$.(I can […]

Calculate rectangle vertices

I have a programming problem and I’ve forgotten some of my math rectangle formulas. I have to make a program that creates a rectangle that isn’t parallel to the X and Y axis, I assume this means a diagonal rectangle in the first quadrant of a Cartesian graph. My inputs are the following: coordinates of […]

Composition of two polynomials

How’s to make the composition of two polynomials? According to this page: If $ P = (x^3 + x) $, $ Q = (x^2 + 1) $ then, $ P\circ Q = P\circ (x^2 + 1) = (x^2 + 1)^3 + (x^2 + 1) = x^6 + 3 x^4 + 4 x^2 + 2 $ […]

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k \text{ divides } n\\ 0&\text{otherwise} \end{cases}\\ \frac1n\sum_{j=0}^{n-1}\sin\left(\frac{2\pi jk}{n}\right)&=0 \end{align*}$$

Solving inequalities comparing $f(x)$ to $0$ where $f$ is an elementary function

Any inequality comparing elementary functions can be rearranged to compare some elementary function $f$ to $0$. What is the best way to approach, in general, solving such inequalities at the precalculus level?

A tricky logarithms problem?

$ \log_{4n} 40 \sqrt{3} \ = \ \log_{3n} 45$. Find $n^3$. Any hints? Thanks!

Find all reals $a, b$ for which $a^b$ is also real

The title is pretty much clear, but here is a more precise formulation: Find all pairs $(a,b)\in\mathbb{R^2}$ for which $a^b$ is also real. I used a CAS to solve the problem and it says that the solution is $$(a=0\land b>0)\lor \left(c_1\in \mathbb{Z}\land a<0\land b=c_1\right)\lor a>0$$ But i think the correct answer is $$(a=0\land b>0)\lor \left(c_1\in […]

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$?

This question already has an answer here: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ 5 answers How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? [duplicate] 3 answers

Help with understanding a proof that $f$ is bounded on $$ (Spivak)

I need help on the proof of Theorem 7-2 in Spivak: If $f$ is continous on $[a,b]$, then $f$ is bounded above on $[a,b]$. So, the proof starts with this: Let $$A= \{x:a\le x \le b \text{ and } f \text{ is bounded above on } [a,x]\}$$ The author then went on to prove that […]