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.

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5
votes
2answers
633 views

Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
1
vote
2answers
38 views

Algebra Logical Pythagorean theorem help

A wire is attached to the top of a pole. The pole is 2 feet shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 feet less than the length of the wire. ...
2
votes
2answers
105 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
0
votes
2answers
87 views

A boy's father is 25 years older than him. The sum of their ages is 31. How old is the boy?

Q.) A boy is $y$ years old. His father is 25 years older than he is. The sum of their ages is 31. How old is he? In class we wrote the answer as $\displaystyle 2y+25=31 \rightarrow 2y=31-25=6 ...
2
votes
2answers
100 views

Finding prime solutions to $100q+80 = p^3 + q^2$

Finding prime solutions to $100q+80 = p^3 + q^2$ Does them being prime imply some patterns on division modulo 3 or some other integer? How is this done?
1
vote
3answers
31 views

Remainders questions help

If we divide a number by 3, 4 ,5 , 6 , we have the remainders 2, 3 , 4 , 5. Is there any way to get a pattern without guessing so many numbers and checking by 3, 4 ,5 ,6?
1
vote
1answer
31 views

How to compute $ \prod_0^n( 1- { 2 \over (2+k)(3+k)}= $?

I have spent quite some time to solve this question, before I asked Wolfram Alpha and got this: $$ \prod_0^n \left(1- {2\over(2+k)(3+k)}\right) = { n+4 \over 3(n+2)}. $$ Now that I know that this ...
3
votes
3answers
64 views

Logarithmic inequality for a>1

Is $\log_{\sqrt a}(a+1)+\log_{a+1}\sqrt a\ge \sqrt6$ always true for $a>1$? What is the approach? Do we check the first a's and then form a induction hypothesis?
1
vote
1answer
49 views

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$ I've got that $x,y,z$ to the 52nd power are congruent to 1 modulo 53 from Fermat's. How is it continued? Help would be ...
3
votes
1answer
63 views

Arithmetic progression with common difference 2061

If there are 30 consequent members of an arithmetic progression with CD of 2061, show that among them are at most 20 squares of natural numbers. I wrote out $a_1$ through $a_{30}$ and tried to find ...
3
votes
2answers
44 views

Show that if x,y are and $ x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0 $ then $x \geq -1/6 $

Show that if x,y are real and $ x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0 $ then $x \geq -1/6 $ So far I've tried factoring $x^2$ and throwing the 8 on the LHS, but can't get to the needed result. Help would ...
1
vote
2answers
49 views

Some help with sin and cos

I'm having trouble to understand the following equalities in these two equations, i.e. how to apply the addition formulas. Firstly: $$ \frac {1- \frac {sin^2(\frac x2)} {cos^2(\frac x2)}} {1+ \frac ...
0
votes
1answer
29 views

Combining two liquids with different weights to achieve a desired volume and weight

I have two liquids - water and alcohol, each liquid has a different mass Water weighs 1 gram per ML Alcohol weighs 0.5 gram per ML (just for the sake of the example) I wish to combine these ...
-1
votes
1answer
24 views

How many milliliters of liquid to fill [duplicate]

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more millilitres of liquid need to be added in order to fill the ...
0
votes
1answer
41 views

Simplify the algebraic expression

Can someone please explain to me how the algebraic expression in the picture is simplified. To be more specific, how (1) becomes (2). $3x^2(6x-4)^4 + x^3(6\times 4\times (6x-4)^3)$ ...
1
vote
4answers
177 views

How to find the 4th degree polynomial with given values at $0,1,2,3,4$?

Determine a fourth degree polynomial p that has $p(0), p(1), p(2), p(3), p(4)$ equal to $7, 1, 3, 1, 7$, respectively. Using my ideas, I first write out the points on the polynomial as $(0,7), (1, ...
1
vote
2answers
57 views

Find all almost lower bounds and almost upper bounds of $\{\frac 1n: n\in \Bbb N\}$

A number $x$ is called an almost upper bound for $A$ if there are only finitely many numbers $y$ in $A$ with $y\ge x$. An almost lower bound is defined similarly. (a) Find all almost lower ...
0
votes
4answers
59 views

How to solve this inequality? $\sqrt x\geqslant x-6.$

How to solve this inequality? $$\sqrt x\geqslant x-6.$$ My answer is $[4,9]$, but it must be $[0,9]$, I don't understand what's wrong. Could you give me solution?
1
vote
1answer
49 views

Strong Induction Proof / Algebra

Alright, I pretty much have the proof done, now just trying to do the algebra on it. This is the question... The information I have is: $$a_k = C_1 r^k + C_2 s^k$$ $$a_{k-1} = C_1 r^{k-1} + C_2 ...
1
vote
1answer
36 views

Some algebra trouble

How do I show that $$ \frac{sa_0-a_1}{s-r} r +\frac{a_1-ra_0}{s-r} s $$ equals $a_1$?
0
votes
1answer
50 views

Issue on proving quadratic formula

I have come across a stage of the proof: $$ \left(x+\frac b{2a}\right)^2=\frac{b^2-4ac}{4a^2}$$ How does $\left(x+\frac b{2a}\right)^2$ not equal $\pm x\pm \frac b{2a}$ when taking the square root?
1
vote
6answers
61 views

How to determine the derivative of $ f $ at $ x=2$ by looking at the graph only?

How to determine the derivative of $ f $ at $ x=2$ (i.e., $ f^\prime(2) $) by looking at the graph only ? I am well aware of the theory of the derivative and how to compute it. But how can I ...
1
vote
4answers
824 views

Solved to be 7 after arithmetic

I recently made a blunder while trying to explain a question asked to me in an interview, The question was Think of $X$ Add $X$ to itself ($X+X = y$) Times the result by $3$ ($y\times 3 = z$) ...
2
votes
3answers
80 views

Finding all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors.

Find all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors. I've tried checking the first 6-7 $n$'s on wolframalpha, but I don't see any patterns for even nor odd $n$'s. At first ...
1
vote
2answers
46 views

Proof by induction of sum

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 40. Exercise 10. Prove by induction, that for $n\ge1$ we have ...
3
votes
2answers
64 views

If $f(x) = \sqrt{x}$, what is the domain of $f^4(x)$?

I am unclear if I should consider the function's domain before or after raising it to the power. My textbook gives the following definition of raising a function to a power: By $f^n$, we mean the ...
0
votes
2answers
44 views

How many milliliters to fill cone

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more milliliters of liquid need to be added in order to fill the ...
0
votes
2answers
34 views

Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
0
votes
1answer
33 views

Linear equation with 3 equal signs

Here is my linear equation: Solve for p: $\frac{5}{6} = \frac{n}{72} = \frac{m+n}{84} = \frac{p-m}{120}$ How am I supposed to solve for this 1 variable when there are multiple equal signs and 3 ...
0
votes
5answers
91 views

Solve $\sqrt{3x}+\sqrt{2x}=17$

This is what I did: $$\sqrt{3x}+\sqrt{2x}=17$$ $$\implies\sqrt{3x}+\sqrt{2x}=17$$ $$\implies\sqrt{3}\sqrt{x}+\sqrt{2}\sqrt{x}=17$$ $$\implies\sqrt{x}(\sqrt{3}+\sqrt{2})=17$$ $$\implies ...
0
votes
2answers
55 views

Two ships leaving a port at different times and different speeds. When do they meet?

Can someone please show me the working out to this word problem I have the answer but have no clue how to do the working out. At noon ship A leaves port steaming at 8 knots 2 hours later ship B ...
-1
votes
1answer
19 views

divide clock into halfs

John has special clocks one hands do 1 turn per minute, second do 1 turn per 3 minutes and third do 1 turn per 15 minutes. how many times and when the first divide clockface into three equal parts in ...
-1
votes
5answers
78 views

Prove that for all positive integers $x$, $\left\lfloor \frac{x^2 +2x + 2}{4}\right\rfloor =\left\lfloor \frac{x^2 + 2x + 1}{4}\right\rfloor$.

Title says it all, basically. I believe it to be true that $$\left\lfloor \dfrac{x^2 + 2x + 2}{4} \right\rfloor=\left\lfloor \dfrac{x^2 + 2x + 1}{4} \right\rfloor$$ for all positive integers $x$. I ...
1
vote
1answer
41 views

Trouble understanding factorial algebra

I am having trouble understanding some of the algebraic concepts used here. In fact, the entire thing to me makes sense, except for the second red line. I don't understand how the diagonal swap ...
0
votes
1answer
62 views

Solving Polynomial Equations and Inequalities

The distance, in km, of a ship from its harbour is modeled by the function $d(t)= -3t^3 + 3t^2 + 18t$ where $t$ is the time elapsed in hours since departure from the harbour. a) When does ...
1
vote
3answers
58 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
6
votes
3answers
127 views

Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that: $ 6 \cdot 3.7 = 22.2$ than it is to see that $ 22.2 \div 6 = 3.7 $. Thoughts about the roots of this asymmetry? An analogous ...
2
votes
3answers
52 views

Positive values of $x$ that satisfy the inequality $\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}$

Determine the set of positive values of $x$ that satisfy the inequality $$\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}.$$ My attempt: \begin{align} \frac{-1}{x(x-1)} & >\frac{1}{(x-2)} \\[0.1in] ...
1
vote
1answer
53 views

Mortgage payment calculation without annuty.

I have been asked the following problem by a student of mine and there is a specific method that he requested. A mortgage of $\$450,000$ is loaned for a monthly payment for $30$ years with nominal ...
1
vote
2answers
45 views

Sum of roots of an equation $\sqrt{x-1}+\sqrt{2x-1}=x$

Find the sum of the roots of the equation $\sqrt{x-1}+\sqrt{2x-1}=x$ My attempt: Squaring the equation: $(x-1)+(2x-1) +2\sqrt{(x-1)(2x-1)}=x^2$ $\implies x^2-3x+2=2\sqrt{(x-1)(2x-1)} $ $\implies ...
0
votes
0answers
24 views

Algebraic problem

Let $T:R\to R$ be given as $Tx=x^3-1$ and $d(x,y)=\left|x-y\right|$. Then is it possible to find $\lambda\in (0,1)$ and $L\geq0$ such that $$d(Tx,Ty)\leq \lambda M(x,y)+LN(x,y)$$ for all $x,y\in R$, ...
2
votes
0answers
34 views

Solve $x/(4x^2+1) = \tan(6x)$ for $x$

$$ \frac{x}{4x^2+1} = \tan(6x) $$ Can this equation be solved for $x$ algebraically and can I get exact answer for this question? Or do I have to approximate it?
0
votes
5answers
62 views

How do I solve this inequality $\frac{c-1}{\sqrt{c}}<2$?

How do I solve this inequality $\frac{c-1}{\sqrt{c}}<2$? Wolfram Alpha says that it is $0<c<3+2\sqrt{2}$ but my brain is just not letting me see how to get there.
0
votes
3answers
59 views

Remainder Theorem, solve for K

For what values of $k$ does the function $f(x) = x^3 + 6x^2 + kx – 4$ give the same remainder when divided by $(x-1)$ and $(x + 2)$?
1
vote
2answers
58 views

Quadratic equation $4x^2+4x=7$ using quadratic formula

Solve using quadratic formula. $4x^2+4x=7$ So $4x^2+4x-7=0$ $A=4$ $b=4$ $c=-7$ $$x=\frac{-4\pm\sqrt{(4)^2-4(4)(-7)}}{2(4)}=\frac{-4\pm\sqrt{16+112}}{8}=\frac{-4\pm\sqrt{128}}{8}$$ What's next?
0
votes
2answers
74 views

Solving a system of two cubic equations

I'm trying to solve a system of two cubic equations with two variables x and y. The original problem was to solve the equation $z^3=-4i \overline{z}$. I know how to solve it using polar form. Now I ...
0
votes
3answers
114 views

Find numerical value without using a calculator [closed]

4^n+4^n+1/4^n-2 Should I leave it as 4 or change it to 2^2. I'm unable to get rid off all variables. Please help.
0
votes
2answers
46 views

Gelfand trigonometry question

If we start with a lemma that states that when $ a^2+b^2=1$ there exists an angle $ \theta $ such that $ a=\cos\theta $ and $ b=\sin\theta$ Suppose that $\alpha$ is some angle if ...
1
vote
3answers
80 views

Using IVT prove that a polynomial of even degree has atleast two real roots if $a_n a_0 \lt 0$

$P(x) = a_n x^{n} + a_{n-1}x^{n-1} + \cdots +a_1x+ a_0$ $n = 2k$ Show that $P(x)$ has at least two real roots if $a_na_0 \lt 0$ I think I need to find some interval of length $|N|$ in which the ...
6
votes
2answers
173 views

how find all the zeroes of the polynomial

Find all the zeroes of the polynomial $f(x) = 2x^7 - 17x^6 -45x^5 +390x^4 + 28x^3 + 1832x^2 +960x$ this is my try $f(x)= 2x^7 - 16x^6-x^6-42x^5-3x^5+390x^4 + 28x^3 + 1820x^2+12x^2 +960x$ ...