Tag: algebra-precalculus

algebra precalculus – Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

The Question : 108 people think this question is useful $x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the

algebra precalculus – If squaring a number means multiplying that number with itself then shouldn’t taking square root of a number mean to divide a number by itself?

The Question : 116 people think this question is useful If squaring a number means multiplying that number with itself then shouldn’t taking square root of a number mean to divide a number by itself? For example the square of $2$ is $2^2=2 \cdot 2=4 $ . But square root of $2$ is not $\frac{2}{2}=1$

algebra precalculus – Division by $0$

The Question : 126 people think this question is useful I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions. $\dfrac {x}{0}$ is Impossible ( If it’s impossible it can’t have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only

sequences and series – Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$

The Question : 126 people think this question is useful Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some values of $x$ for which the above formula is invalid? What about if we take only a

sequences and series – Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

The Question : 164 people think this question is useful I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ using mathematical induction. I’m interested if there’s an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs. The Question Comments : math.stackexchange.com/questions/61798/… Look at this