Articles of algebra precalculus

Arithmetic progression

I’m studying for a test, and I am having a hard time with this particular exercise. The first member is equal to 7 and the fifth member is equal to 59. How many members should be taken in to the sequence that it would amount to 24,217? So far I have found out that d=13, […]

Is $\sqrt{x^2}$ always $\pm x?$

I am wondering if this holds in every single case: $$\sqrt{x^2} = \pm x$$ Specifically in this case: $$\sqrt{\left(\frac{1}{4}\right)^2}$$ In this one we know that the number is positive before squaring, so after removing the square and root shouldn’t we just have: $$\frac{1}{4}$$ Also in a case such as this: $$\sqrt{(\sqrt{576})-8}$$ we have $$\sqrt{\pm24-8}$$ which […]

An Algebraic Proof that $|y^3 – x^3| \ge |(y – x)|^3/4 $

I can prove this using calculus, but not by simple algebra: can anyone help ? Calculus Proof: Fix the separation of $x$ and $y$ so that $y = x + d$ with $d>0$ ($ \implies y > x \implies y^3 > x^3$) and now consider $ f(x) = y^3 – x^3 = (x+d)^3 – x^3$. […]

Manipulating Algebraic Expression

$a + b + c = 7$ and $\dfrac{1}{a+b} + \dfrac{1}{b+c} + \dfrac{1}{c+a} = \dfrac{7}{10}$. Find the value of $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b}$. I algebraically manipulated the second equation to get: $\dfrac{(b+c)(c+a) + (a+b)(c+a) + (a+b)(b+c)}{(a+b)(b+c)(c+a)} = \dfrac{7}{10}$ $\dfrac{bc+ab+c^2+ac+a^2+bc+ba+ab+ac+b^2+bc}{(a+b)(b+c)(c+a)} = \dfrac{7}{10}$ $\dfrac{(a+b+c)^2}{(a+b)(b+c)(c+a)} = \dfrac{7}{10}$ $\dfrac{7^2}{(a+b)(b+c)(c+a)} = \dfrac{7}{10}$ $(a+b)(b+c)(c+a) = 70$ I’m stuck after this.

Calculation of $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot 3^m\right)}$

This question already has an answer here: I need help with a double sum [closed] 1 answer Double sum trouble 2 answers

$ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $

If $ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $ Dont know what kind of series is this. Please educate. How to do such problems?

Solving Cubic when There are Known to be 3 Real Roots

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ is positive. The equations to find the 3 roots are: $x_1 = -\frac{1}{3a}(b + C + \frac{\Delta_0}{C})$ $x_2 = […]

Where is the order of the variables inside the parentheses coming from?

I’m reading Sawyer’s Prelude to Mathematics, here: I can’t understand what’s the meaning and application of “condition” here. Also when he gives the example on the cubic equation, stating that the condition is: $$(bc-ad)^2-4(ac-b^2)(bd-c^2)=0$$ I can understand that it is $b^2-4ac=0$ (I hope I’m right with this), I just have no idea on where is […]

2011 AIME Problem 12, probability round table

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. I have been searching around math stackexchange […]

Trigonometric equation $\sec(3\theta/2) = -2$ – brain dead

Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don’t know what to do afterwards, the answer should be a huge list of $\theta$s, which I cannot seem to get.