0
votes
1answer
63 views

what is the coefficient of following expression

what is the co-efficient of $x^{50}$ in the expansion of $$\frac{1}{(1-x^{1.7})(1-x^{1.8})(1-x^{2.6})(1-x^{3.0})(1-x^{4.0})(1-x^{6.7})(1-x^{7.5})(1-x^{8.2})}$$ can you please explain me the logic
-1
votes
2answers
317 views

What is the efficient way to calculate number of divisors of N that are divisible by 2?. [closed]

For example if a number is given let say 8 then its factors are 1,2,4,8 hence total numbers of divisors which are divisible by 2 are (2,4,8) that is 3.
0
votes
0answers
21 views

How to find a certain uppper bound (see details)?

What would be the most efficient way to find this upper bound? Given natural number n and a natural number d < n, find the ...
0
votes
1answer
47 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
4
votes
5answers
326 views

Find a closed expression for a formula including summation

Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...
-2
votes
1answer
44 views

Comparing floor and ceiling fractions

Is the following true for all integers x>1: $\lfloor{\frac{2x}{3}}\rfloor \geq \lceil \frac{x}{2}\rceil$
0
votes
2answers
35 views

Intro to Discrete Math: compound interest calculation

The following is from an intro to discrete mathematics page. It's on compound interest. http://www.cs.odu.edu/~toida/nerzic/content/intro2discrete/intro2discrete.html[1] Scroll to the part with the ...
3
votes
3answers
77 views

Values of $n$ for which $\lfloor 2 x\rfloor +\lfloor 4 x\rfloor +\lfloor 8 x\rfloor +\lfloor 20 x\rfloor =n$ has a solution

$$\lfloor 2 x\rfloor +\lfloor 4 x\rfloor +\lfloor 8 x\rfloor +\lfloor 20 x\rfloor =n$$ How would you find the values of $n$ for which the equation has a solution under the condition that $n \leq ...
0
votes
2answers
49 views

show that $\det(A)=0$ in this case

(a) Let $x$ and $y$ be $n\times 1$ matrices, $n \ge 1$, and let $A=xy^T$. Show that $\det(A)=0$. (b) Explain why the statment in part (a) is false if $n=1$.
5
votes
1answer
67 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 ...
-1
votes
1answer
50 views

Password Strength

If passwords of exactly $8$-characters are used, and the character set consists of just lower-case alpha (a-z), how many passwords are possible? Expand the character set to include (A-Z), (a-z), ...
4
votes
2answers
70 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
41 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
49 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
99 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
428 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
45 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
84 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
215 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
71 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
34 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
105 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
47 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
154 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
101 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
56 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
52 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
63 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
57 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
53 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
53 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
303 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
361 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
141 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
62 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
384 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
374 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
81 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
1k 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
53 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
80 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
1k 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
287 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
votes
2answers
106 views

How many 2007 digit number exists, where each two digit number is made up of two neighbour digits and can be divided by 17 or 23?

I know that there exists nine 2007 digit number where each two digit number is made up of two neighbor digits, those numbers are: $$12345678901234....90123456789012345678 $$ ...
1
vote
3answers
65 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
88 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 ...