5
votes
0answers
45 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
4
votes
2answers
60 views

Solving $x^{\left\lfloor x \right\rfloor}\; =\; 2014$

$x^{\left\lfloor x \right\rfloor}\; =\; 2014$ Mathematica gives that there are no solutions, but how do you actually come to the conclusion that there exists no solution to this equation?
1
vote
2answers
39 views

Find a recurrence for in , the number of integer compositions of n which only have 1s and 2s as parts.

Find a recurrence for $$i_n$$ the number of integer compositions of $n$ which only have $1$s and $2$s as parts. How do you approach this problem?
0
votes
1answer
36 views

Logical form of this statement?

In logical form, how would you express : Take any two fractions, add them together, and the result will be an integer
0
votes
2answers
29 views

What's the maximum deviation from loan amortization

Suppose you have a loan with principle P and fixed interest rate i compounded daily. Suppose you make fixed payments every month, but not on the same day. The only constraint is that you make every ...
0
votes
5answers
89 views

Prove that $\left\lfloor \lfloor x/2\rfloor/2 \right\rfloor=\lfloor x/4\rfloor$ for all $x$. [duplicate]

This I approached the problem. I let $x = n + e$ where $n$ is an integer and $e$ is a decimal less than $1$ but not less than $0$. I substituted that into the equation to get $\left\lfloor \lfloor ...
0
votes
2answers
95 views

How to scale numbers from one range to another range?

I'm stuck in a problem of mapping numbers from one range to another. I want to calculate popularity of a web page based on the number of page hits on a scale of 10. The problem is total number of web ...
0
votes
2answers
35 views

Random number x probability in interval [0,10]

So lets say $x=10*rand()$, so x will be equal to a random number on the interval [0,10]. So what is the probability of $2x+1$ rounding to $5$. So one can come up with $4.5<=2x+1<5.5$. Is that ...
1
vote
1answer
36 views

Chain and sprockets

Consider a sprocket has $s$ teeth, and a chain has $h$ holes. Think about a bicycle chain and the sprocket wheel in the back of the chain. Sprockets are numbered $0 \ldots s-1$ and holes are numbers ...
2
votes
3answers
75 views

More accurate estimation of mathematical constant $e$

Very often in books and also on Wikipedia we can find that: $$e \approx \left(1+\frac{1}{n}\right)^n$$ but I want more accurate estimation, it means instead using $\approx$ I wonder if I can use ...
1
vote
3answers
184 views

could not able to understand Project Euler 18. “Maximum path sum I”

According to question, By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23. ...
0
votes
1answer
63 views

equation of the tangent line to the given curve

Find an equation of the tangent line to the given curve at the specified points: $F(t)=(t+1, t^2, 2t - 1)$ at the point $(2,1,1)$ Teacher said the we should start like that: $r(t) = f(t) + t ...
-1
votes
1answer
32 views

Which equation has a solution $(x,y)$ i which both $x$ and $y$ are integers?

Which equation has a solution $(x,y)$ in which both $x$ and $y$ are integers? $12x + 9y = 16$ $32 x + 80y = 27$ $42x + 56y = -28$ $20x + 90y = 105$ Do we have to use discriminant ($b^2 - 4ac)$ ...
1
vote
1answer
96 views

For what values of m does the equation 35530x + 355y = m have integer solutions?

For what values of $m$ does the equation $35530x + 355y = m$ have integer solutions? (only find the $m$'s for which solutions exist)
0
votes
1answer
44 views

help on manipulating this algebraic expression

So I have something like: $\frac {k!}{(k-3)!3!}$ I'm going to add $\frac 12k(k-1)$ to this, and I want to obtain $\frac {(k+1)!}{(k-2)!3!}$ as the result. I'm having trouble with this since I need ...
0
votes
3answers
137 views

Why does $0,\bar{9}$ equal $1$? [duplicate]

I am finding hard to understand why $0,99999..... = 1$ I have the following proof: Let $x$ be $0,9999...$ then $10x = 9,999...$ So $10x - x = 9,999 - 0,9999$ $9x = 9 \rightarrow x = 1$ From a ...
0
votes
2answers
87 views

Solve discrete Math Problem using abstract algebra, postage problem?

The question I am looking at is not very hard: Determine which amounts of postage can be written with $5$ and $6$ cent stamps. To determine the amount, use a brute force way to solve it. Counting ...
2
votes
2answers
54 views

Is this equation to prove that $aRb \iff a^2 - b^2 = 1$ is antisymmetric correct?

Over $\mathbb{R}$, $aRb \iff a^2 - b^2 = 1$. I tried determining if it was antisymmetric. I seem to have done it, but while doing the equation, I stumbled upon a scenario that always made me ...
1
vote
2answers
48 views

Finding Integers to Sum to 1200

$125x_1+25x_2+5x_3+x_4=1200$ Find all ordered pairs of integers such that $0 \leq x_1,x_2,x_3,x_4 \leq 25$. Is there any systematic way of approaching this? Or do you need to solve by ...
1
vote
0answers
54 views

How to use an exponent that contains a variable

I am trying to understand a problem that uses mathematical induction to prove the validity of a statement. This is how one section moves to another: $$ 2k + 3 = 2^{k + 1} $$ $$ 2k + 3 = (2k + 1) + 2 ...
-1
votes
1answer
55 views

I am not how they got characteristic equation from the given equation.

![can someone tell me they got characteristic equation from the given recursive equation.][1] i know how to do the rest of problem but getting characteristic equation stopped me. The recurrence is ...
0
votes
2answers
50 views

Limits of logarithms with different bases.

Although this is a Discrete Structures problem, I am having trouble solving the pre-calculus portion of this problem. The exercise gives us a $f(x)$ and a $g(x)$, and to figure out the asymptotic ...
2
votes
2answers
51 views

Finding Polynomials to Satisfy a Condition

I need to find polynomials $x(n), y(n)$ s.t. $x(n)(n^{2}+n+1)+y(n)(n^{2}+1)=1$, $\forall n \in \mathbb{Z}$. I tried distributing it out: $x(n)n^{2} + x(n)n + x(n) + y(n)n^{2}+y(n)=1$. I understand ...
1
vote
1answer
266 views

How many $n$-disk legal configurations are there for the Tower of Hanoi?

This question comes from this homework assignment from ECS20 at UC Davis. How many $n$-disk legal configurations are there for the Tower of Hanoi? A "legal configuration" means that no disk is ...
0
votes
1answer
298 views

How to prove “If $R$ is transitive, then $R^n$ is transitive.”?

I can understand $R^n$ is $R$'s subset, but I can't understand why $R^n$ is transitive,too. I used mathematical induction: Basis step: Let $n = 2$. If $a R^2 b$, $b R^2 c$, I need to prove $a R^2 c$. ...
1
vote
2answers
122 views

Equation for determining a car's fuel consumption as well as cost

Purchase price: 24000 Avg km/year: 40000 L/100 km: 5.3 Price of gas (per L):1.30 I was wondering what the formula is to find out how much litres of gas the car would consume as well as the cost of ...
0
votes
3answers
74 views

Discrete Math need help please

Need help with this please I have no clue
2
votes
1answer
59 views

How to find the base $n$ such that $2_{n}^{12_n}=2_{10}^{6_{10}}\cdot 5_{10}$?

I have met the following problem: How to find the base $n$ such that $2_{n}^{12_n}=2_{\small10}^{6_{\small 10}}\cdot 5_{\small 10}$? And until now, I have no idea of how to solve it. I could try ...
4
votes
3answers
370 views

17! mod 13, How do I do this without a calculator

So I know $$17! = 17 \times16\times15...\times1$$ So I was thinking maybe go $$17mod(13)\equiv4 \space \space and \space 16mod(13)\equiv3 ...$$ add all that together but that is too much work so I ...
3
votes
1answer
268 views

Concrete Mathematics Prerequisite Question

I've been very interested in the book Concrete Mathematics (Graham,Knuth,Patashnik) and I've been reading it for the past few weeks. I'm at the chapter about Sums (Chapter 2), specificaly, the lesson ...
0
votes
1answer
79 views

Why is this summation formula wrong?

This is the alternate form of the summation formula: $$ \sum^{n}_{k=0} a(c)^k = \frac{ac^{n+1} - a}{c - 1} $$ so why is this wrong? $$ \sum^{n}_{k=0} (-\frac{1}{2})^k = \frac{(-\frac{1}{2})^{n+1} - ...
1
vote
1answer
772 views

Suppose a,b are real numbers, if a is rational and ab is irrational, then b is irrational (Is my solution correct?)

Suppose $a,b$ are real numbers, if $a$ is rational and $ab$ is irrational, then $b$ is irrational. Solution: Proof by contraposition $$b = \frac{p}{q}$$ $$ a = \frac{j}{k}$$ where $p,q,j,k$ are ...
3
votes
1answer
51 views

How do I apply partial fraction expansion on $\dfrac{K}{(a+bz^{-1})(x+yz)}$?

I want to apply partial fraction expansion on $\dfrac{K}{(a+bz^{-1})(x+yz)}$. I'm not able to do it in the standard way, because one term has $z^{-1}$ term and the other has $z$. What is the approach ...
0
votes
3answers
70 views

Recurrence relation of two next terms

For the recurrence relation, $a_{n+2}=3a_{n+1}-2a_n$ with $a_0=2$ and $a_1=3$, compute the first six terms of the sequence and derive a closed form formula for this sequence. So I'm totally lost with ...
2
votes
3answers
978 views

Counting 1:1 and onto functions

I'm faced with the following questions: 1) How many functions are there from a set of size 3 to a set of size 5? How many of them are 1-to-1? 2) How many functions are there from a set of ...
1
vote
2answers
233 views

Strong Induction: Prove provided recurrence relation $a_n$ is odd.

I'm not sure if we're allowed to post pictures but I thought it would be easier to read and I didn't see anything in the rules about it. It's question 1. Section 5.4 This question: Here is the ...
1
vote
3answers
63 views

Prove Base case $P(0)$ with three variables.

I'm studying for a mathematical induction test tomorrow, and I have a practice question: Use mathematical induction to prove that if $a$ and $b$ are integers with $a \equiv b \pmod m$ then $a^k ...
0
votes
2answers
86 views

$n\le 20$ What is the maximum number of digits representing $n$ with base $4$?

$n\le 20$. What is the maximum number of digits representing $n$ with base $4$? This is a question on a sample test that I'm studying for. My prof hasn't gone over it in class, can someone direct ...
0
votes
1answer
40 views

Help with $7 · 7^{k+2} + 64 · 8^{2k+1}$=$7(7^{k+2} + 8^{2k+1}) + 57 · 8^{2k+1}.$

Could someone please explain why 7 is being subtracted from 64 in: $7 · 7^{k+2} + 64 · 8^{2k+1}$ to make $7(7^{k+2} + 8^{2k+1}) + 57 · 8^{2k+1}.$ Also, how does $7^{k+2}$ get factored by 7 to ...
1
vote
1answer
104 views

How does $(k^3 + 3k^2 + 3k + 1) − (k + 1)=(k^3 − k) + 3(k^2 + k)$?

More stuff from my textbook that I'm not quite understanding, help is appreciated. I'll be trying to figure it out and updating my question as I wait for answers. I understand how $3k^2+3k$ factors ...
4
votes
6answers
111 views

Process for $(k+1)^3$?

I've mentioned in previous questions that I have a hard time with simplifying algebraic equations. For this equation, I assumed we would first put the equation into easier to read form: ...
8
votes
2answers
267 views

How does $2^{k+1} = 2 \times 2^k$?

I ask only because my textbook infers this in an example. Where should I go to learn more about this? I'm trying to learn mathematics by Induction but my knowledge of simplifying algebraic equations ...
1
vote
2answers
38 views

How does this prove that P(k) of every k is true?

This is an example from my textbook. I'm very rusty with simplifying algebraic expression so i hope you'll forgive me for that. The textbook says there are two rules to Mathematical Induction: 1) We ...
0
votes
1answer
374 views

What Precalculus knowledge is required before learning Discrete Math Computer Science topics?

Below I've listed the chapters from a Precalculus book as well as the author recommended Computer Science chapters from a Discrete Mathematics book. Although these chapters are from two specific ...
5
votes
2answers
235 views

Evaluating complicated sum

Evaluate for a fixed $m\neq 1$ ( $m\in \mathbb{N}$ ) $$\sum _{k=1}^{n}\left[\left( \sum _{i=1}^{k}i^{2}\right) \left(\sum _{k_{1}+k_{2}+...+k_{m}=k}\dfrac {\left( k_{1}+k_{2}+\ldots +k_{m}\right) ...
3
votes
1answer
153 views

inequality with sum of powers

How to prove the following inequality: $$\forall n\geqslant 4:\dfrac {3^{n}+4^{n}+\cdots +\left( n+2\right) ^{n}} {\left( n+3\right) ^{n}} < 1$$
2
votes
4answers
877 views

Sum to closed form?

Is there a general method for removing a sum from an expression to produce a closed form? For example I needed to "unroll" the following expression in a recent programming competition (as $k_1$ and ...
1
vote
4answers
348 views

Proof by Contradiction Problem Where do i start

Prove the following: There are no rational number solutions to the equation $x^3 +x+ 1$ = 0, i.e. no solution can be written as a ratio a/b where a, b ∈ N (you can always consider a/b to be reduced to ...
0
votes
2answers
196 views

Simplifying exponents, multiplication, and addition

How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.
1
vote
5answers
424 views

General solution using Euclidean Algorithm

I was able to come up with the integer solution that they also have in the textbook using the same method they used but I am really puzzled how they come up with a solution for all the possible ...