linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics
19
votes
3answers
479 views
Why everytime the final number comes the same?
I have come across an interesting puzzle.
Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with
$\text{Number}_{new} = xy/(x + y)$
OR
$\text{Number}_{new}= x + y + xy$
...
9
votes
5answers
1k views
No radical in the denominator — why?
Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
9
votes
2answers
574 views
Is this a known algebraic identity?
In the course of analyzing a certain Markov chain, I once had to prove the following algebraic identity.
Is there a slick or known proof?
For $n$-tuples $(x_1,x_2,\dots, x_n)$ of positive real ...
6
votes
11answers
5k views
Derivation of the formula for the vertex of a Parabola
I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form
$y = a x^2 + b x + c$
My teacher gave me the formula:
$x = -\frac{b}{2a}$
as the $x$ ...
4
votes
2answers
304 views
How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?
This sum is difficult. How can I compute it, without using calculus?
$$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$
If someone can explain some technique to do it, I'd appreciate it.
Or advice using ...
4
votes
6answers
332 views
The existence of partial fraction decompositions
I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it ...
3
votes
5answers
283 views
How to “Re-write completing the square”: $x^2+x+1$
The exercise asks to "Re-write completing the square": $$x^2+x+1$$
The answer is: $$(x+\frac{1}{2})^2+\frac{3}{4}$$
I don't even understand what it means with "Re-write completing the square"..
...
3
votes
3answers
645 views
How to find the sum of the following series
How can I find the sum of the following series?
$$
\sum_{n=0}^{+\infty}\frac{n^2}{2^n}
$$
I know that it converges, and Wolfram Alpha tells me that its sum is 6 .
Which technique should I use to ...
2
votes
4answers
402 views
Steps to solve this system of equations
I want to solve this system of equations, I have been out of Maths for a long !!
$$\sqrt{x} + y = 7$$
$$\sqrt{y} + x =11.$$
Just wondering easiest step to find values for $x$ and $y$ from the above ...
0
votes
1answer
76 views
What is the minimum number of blocks to build this?
A rectangular solid is built using $N$ cubes of a side length of 1cm. When viewed from such an angle such that only 3 of the sides of the rectangular solid are visible, there are 231 $cm^2$ of area ...
7
votes
4answers
274 views
Algorithms for “solving” $\sqrt{2}$
The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
5
votes
4answers
818 views
Geometric mean never exceeds arithmetic mean
This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.)
The geometric ...
4
votes
5answers
1k views
How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$?
I'm doing exercise on discrete mathematics and I'm stuck with question:
If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f ...
3
votes
2answers
217 views
If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$?
If $2 x^4 + x^3 - 11 x^2 + x + 2 = 0$, then find the value of $x + \frac{1}{x}$ ?
I would be very grateful if somebody show me how to factor this polynomial by hand, as of now I have used to ...
3
votes
3answers
276 views
Question about solving absolute values
I solved the following problem from by book, but the answer of this problem at the end of book is x < = 3 Please tell me how can i get this answer.
2
votes
3answers
209 views
Finding the error in a proof
I have a "proof" that has an error in it and my goal is to figure out what this error is. The proof:
If x = y, then
$$
\begin{eqnarray}
x^2 &=& xy \nonumber \\
x^2 - y^2 &=& xy - ...
2
votes
4answers
445 views
Solving a quadratic inequality
I am solving the following inequality, please look at it and tell me am i write or not because this is an example in Howard Anton's book, i solved it on my own as given below but in the book it is ...
1
vote
2answers
94 views
How do I find the delta analytically for $f(x)$ with a degree other than $1$
So I know how to find the $\delta$ in $f(x)$ that can be factored into a degree of $1$, or I can solve it by solving $L + \epsilon = f(x) $ then finding the distance from $a$. But how do I find the ...
1
vote
4answers
269 views
Power summation of $n^3$ or higher [duplicate]
Possible Duplicate:
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
If I want to find the formula for $$\sum_{k=1}^n k^2$$
I would do the next: $$(n+1)^2 = a(n+1)^3 ...
1
vote
1answer
163 views
1
vote
4answers
177 views
the sum of powers of $2$ between $2^0$ and $2^n$
Lately, I was wondering if there exists a closed expression for $2^0+2^1+\cdots+2^n$ for any $n$?
0
votes
2answers
82 views
Problem with Abel summation
Let $$S_n=\sum_{i=1}^{n}\sin k,\quad S_0=0.$$
Then
$$\sum_{k=1}^{n}\frac{\sin k}{k}=\sum_{k=1}^{n}\frac{S_k-S_{k-1}}{k}=\frac{S_n}{n}+\sum_{k=1}^{n-1}\frac{S_k}{k(k+1)}.$$
Could someone please ...
-1
votes
2answers
230 views
How to find all real solution to satisfy this equation without casework or bruteforce?
How to find all real solution to satisfy this equation without casework or bruteforce?
a+b+c=abc
Thanks in advance!
16
votes
16answers
2k views
Interesting calculus problems of medium difficulty?
I would like this to be a community wiki. But, I don't know how to do that here.
I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus ...
15
votes
5answers
403 views
Solving a peculiar system of equations
I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply ...
12
votes
4answers
392 views
How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$
Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
11
votes
2answers
406 views
Given $n! = c$, how to find $n$?
I'm dealing with a time-complexity problem in which I know the running time of an algorithm:
$$t = 1000 \mathrm{ms} .$$
I also know that the algorithm is upper bounded by $O(n!)$.
I want to know ...
7
votes
7answers
1k views
Algebraic Identity $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$
Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$.
So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial ...
13
votes
4answers
201 views
Should the domain of a function be inferred?
It is a common practice to have students of elementary algebra infer the domain of a function as an exercise. I believe this is contrary to the spirit of the definition of a function as a collection ...
9
votes
2answers
166 views
Determining the number $N$
Let $1 = d_1 < d_2 <\cdots< d_k = N$ be all the divisors of $N$ arranged in increasing order. Given that $N=d_1^2+d_2^2+d_3^2+d_4^2$, determine $N$. The divisors include $N$. It seems that ...
11
votes
4answers
465 views
Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$.
Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$
I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? ...
9
votes
2answers
256 views
8
votes
2answers
263 views
Prove the following property of $f(x)$?
Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+...+a_n\sin(nx)|.$$
Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+a_3+....|$ is less than or equal to ...
7
votes
6answers
423 views
What is the result of $\lim\limits_{x \to 0}(1/x - 1/\sin x)$?
Find the limit:
$$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$
I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
6
votes
5answers
475 views
Prove this number fact
Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$.
Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought ...
4
votes
4answers
109 views
Working with proofs help?
I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
4
votes
1answer
243 views
One sum of squares and two Diophantine equations
This question comes from trying to see why 24 is the only non-trivial value of $n$ for which
$$1^2+2^2+3^2+\cdots+n^2$$
is a perfect square.
To this end, let $m,n \in \mathbb N$ be such that ...
3
votes
3answers
492 views
Gaussian proof for the sum of squares?
There is a famous proof of the Sum of integers, supposedly put forward by Gauss.
$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$
$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$
$$S=\frac{n(1+n)}{2}$$
...
3
votes
3answers
1k views
The logic behind the rule of three on this calculation
First,
to understand my question, checkout this one:
Calculating percentages for taxes.
Second,
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final ...
2
votes
3answers
93 views
How to expand $(a_0+a_1x+a_2x^2+…a_nx^n)^2$?
I know you can easily expand $(x+y)^n$ using the binomial expansion. However, is there a simple summation formula for the following expansion?
$$(a_0+a_1x+a_2x^2+...+a_nx^n)^2$$
I found something ...
1
vote
2answers
73 views
Show that equation has no solution in $(0,2\pi)$
Hi I want to show that the equation $2=2 \cos(x)+x \sin(x) $ has no solution in $(0,2 \pi)$. Since it is algebraically impossible to solve this equation for $x$ I wanted to ask you whether one of you ...
1
vote
4answers
391 views
Problems with Inequalities
It seems like I am facing some confusion while handling with inequalities,I was doing some tasks where it is asked to find the interval of the variable,after some steps I deduced the the necessary ...
15
votes
4answers
632 views
Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$ [duplicate]
Possible Duplicate:
How to prove that the number 1!+2!+3!+…+n! is never square?
Show that $\displaystyle\sum\limits_{i=1}^k i!$ is never a perfect square for $k\ge4$
I could prove ...
12
votes
5answers
366 views
How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$
Let $p \in \mathbf{N}$. I don't know how to prove that
$$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$
and
$$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$
...
12
votes
2answers
891 views
What is the best way to solve an equation involving multiple absolute values?
An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases}
ax-b & \text{ if } x\ge \frac{b}{a} \\
b-ax & \text{ if } x< \frac{b}{a}
\end{cases}$, ...
11
votes
8answers
443 views
How to factor quadratic $ax^2+bx+c$?
How do I shorten this? How do I have to think?
$$ x^2 + x - 2$$
The answer is
$$(x+2)(x-1)$$
I don't know how to get to the answer systematically. Could someone explain?
Does anyone have a link to ...
8
votes
2answers
372 views
Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ [duplicate]
Possible Duplicate:
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
How do I prove the following?
$$\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + ...
7
votes
2answers
157 views
Maximum of the difference
What is the maximum value of
$f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$
where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
6
votes
1answer
220 views
$f(x)=x^3+ax^2+bx+c$ has roots $a,b$ and $c$
How many ordered triples of rational numbers $(a,b,c)$ are there such that the cubic polynomial $f(x)=x^3+ax^2+bx+c$ has roots $a,b$ and $c$?
The polynomial is allowed to have repeated roots.
4
votes
3answers
264 views
There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.
This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches.
Please see pg 162-163 on books.google.ca/books?isbn=0387290524
...
