Recent content by IHateFactorial

Given a cube, choose a vertice and draw 2 of the three possible diagonals. What is the measure of the angel between those two diagonals? Proposed answer: We can say that both diagonals touch vertice A, to give it a name. We can also call the endpoints of both diagonals B and C. If we imagine...

Problem: Prove that any positive fraction plus its inverse is greater than or equal to two. Proof: $$\frac{a}{b}+\frac{b}{a}\ge2$$ $$\frac{a^2+b^2}{ab}\ge2$$ $${a^2+b^2}\ge2ab$$ $$a^2+b^2 - 2ab\ge0$$ $$a^2 - 2ab + b^2\ge0$$ $$(a-b)^2\ge0$$ This is true for all a and b: Case 1...

Well, let's look at how this works. Quadratic equations can have either 1, 2, or no zeroes. If it has no real zeroes, the zeroes it DOES have are complex, so that's obviously not it. Let's imagine $$ax^2 + bx + c = 0$$ has one zero, call it $$ \alpha$$ (Cuz it looks pretty). Then that means...

So, there's this problem: $$A = \frac{1}{6}((\log_{2}\left({3})\right)^3 - (\log_{2}\left({6})\right)^3 - (\log_{2}\left({12})\right)^3 - (\log_{2}\left({24})\right)^3)$$ Find $$2^A$$ What I've figured out is that all the logs factorize to 3 + 2 to some power of n...

Let $${a}_{n+1} = \frac{4}{7}{a}_{n} + \frac{3}{7}{a}_{n-1}$$ where a0 = 1, and a1 = 2. Find $$\lim_{{n}\to{\infty}}{a}_{n}$$ Well, seeing as it says that x approaches infinity, the difference between where points an-1, an, and an+1 are plotted on the y-axis is almost insignificant, so we can...

My bad, I didn't include that: The COMPLETE instructions are: Let a and b be real, positive numbers such that their product is one and the sum of their squares is 4. Find the exact value of the expression: $$a^{-3} + b^{-3}$$

I believe the bisecting bit means that, given a quadrilateral ABCD, where AD and BC are the diagonals, say they intersect at point M; they bisect is AM = MD and BM = MD... Unless you already knew that.

Can someone check if this is right? So, having two numbers, a and b, we can say that their product is 1, and the sum of their squares is 4, find the sum of: $$a^{-3} + b^{-3}$$ Well, we have: $$ab = 1$$ $$a^2 + b^2 = 4$$ This means that a and b are reciprocals... Thus: $$a^{-3} + b^{-3} =...

Imagine an 8x8 chess board. In how many ways can 8 queens be placed on the board such that no queen can "eat" any other queen.

If I have a number n and I want to know all the unique ways in which I can use u addends to get that number... How do I do it? For example: If the number is 6 and I want to see how many unique ways I can add up to it (order matters: $$1 + 1 + 2 + 2\ne 2 + 2 + 1 + 1$$) by using 4 addends...

Somewhere I saw that the sum of the infinite arithmetic series $$\sum_{n=1}^{\infty}n = \frac{-1}{12}$$ Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a...

Sorry, my bad. I forgot to take into account the repeated numbers in each unique combination. I.e. 1333 can only be cominated in 4 ways, no 4! ways (they'd be: 1333, 3133, 3313, 3331). I'll fix that in a bit. Factoring in for repeated numbers we have $$ (4 \cdot 4!) + (4 \cdot 4!/2!) + (3...

Well... There are plenty of ways to get 10 by adding 4 numbers from 0-5, let's see them: 0055 1315 2224 0145 1225 2233 0235 1144 2242 0244 1234 0334 1333 (continuing would simply mean repeating the same addition, so we're not going to do that. I BELIEVE that's all of the UNIQUE ways of...

Yes, I have. I do know that the first one simplifies to 1 + 3 + 3 + 1 which is 8, but the generalizing part stumps me. As for the second, I have absolutely NO idea how to prove Pascal's Identity, aside from putting in the variables into their Combination formula. Also, thanks for the...

*sigh* As the title says, they are the bane of my existence... I'd really appreciate it if you guys could help me with these bloody things. 1. Prove that 3C0 + 3C1 + 3C2 + 3C3 = 23 Generalize the formula for any value of r and n such that 0<=r<=n. 2. Prove that n-1Cr + n-1Cr-1 = nCr 3. i) How...