Articles of algebra precalculus

Sum of first n natural numbers proof

I know how to prove this by induction but the text I’m following shows another way to prove it and I guess this way is used again in the future. I’m confused by it. So the expression for first n numbers is: $$\frac{n(n+1)}{2}$$ And this second proof starts out like this. It says since: $$(n+1)^2-n^2=2n+1$$ […]

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I’m really stuck on this one. Tried some stuff involving QM> AM(because the are positive) but can’t derive the needed ,can’t proceed from it.

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$

How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$. I am trying by calculus but don’t know how to use here in this problem. Any idea?

Polynomial with integer coefficients ($f(x)=ax^3+bx^2+cx+d$) with odd $ad$ and even $bc$ implies not all rational roots

Today, I attempted this problem in the ISI admission test for B.Math UG2016 for which I want my solution to be verified (whether it is correct or not) Q: Given $f(x)=ax^3+bx^2+cx+d$ where $a,b,c,d\in\Bbb Z$ and $ad$ is odd and $bc$ is even. Prove that $f(x)$ cannot have all rational roots. My attempted solution: Let us […]

AGM Inequality Proof

I have been stuck on this one for hours. Let $x$, $y$, $z$ be non-negative real numbers. Also we know $x + z \leq 2$. Prove the following: $(x – 2y + z)^2 \geq 4xz – 8y$. Apparently this can be proven with or without AGM, which is $xy \leq \left(\frac{x + y}{2}\right)^2$. This is […]

The number of ones in a binary representation of an integer

Is there any relation that tells whether the number of ones in a binary representation of an integer is an even or an odd number?

Domain of a function

I am confused about this problem: Find the domain of the function, $$f(x)=\frac{x^3-1}{2x^2+5}.$$ I’m guessing it’s all real numbers but the book gives a different answer. The book gave $$(-\infty,-1)\cup (-1,0)\cup (0,\infty)$$ as the answer.

An elegant way to solve $\frac {\sqrt3 – 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2 $

The question is to find $x\in\left(0,\frac{\pi}{2}\right)$: $$\frac {\sqrt3 – 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2 $$ What I did was to take the $\cos x$ fraction to the right and try to simplify ; But it looked very messy and trying to write $\sin x$ in terms of $\cos x$ didn’t […]

Is there a difference between $(x)^{\frac{1}{n}} $ and $\sqrt{x}$?

Is there a difference between $(x)^{\frac{1}{n}}$ and $ \sqrt[n]{x}$ ? I’m confused with this topic. Any ideas or examples ? If $(x)^{\frac{1}{n}} = \sqrt[n]{x}$ Consider $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . Is it the same if I write $ x=\frac{-b \pm(b^2-4ac)^{\frac{1}{2}}}{2a}$ ?

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ $\Rightarrow -2x=W(\dfrac{-1}{3})$ $\therefore x=\dfrac{-1}{2} W(\dfrac{-1}{3})$ But when i went to WolframAlpha, it showed the same result but in the graph: WolframAlpha Graph The curves intersect at a point… And hence there is a second […]