Arithmetic Definition and 474 Threads

Homework Statement Show that if ##0 \leq a < b##, then $$\frac {b^{n + 1} - a^{n + 1}} {b - a} < (n + 1)b^n$$ Homework Equations None that I'm aware of. The Attempt at a Solution Proof (Induction) 1. Basis Case: Suppose ##n = 1##. It follows that: $$\frac {b^{1 + 1} - a^{1 + 1}} {b - a} < (1 +...

Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...

Homework Statement In an AP, sum of first n terms is equal to m and sum of first m terms is equal to n. Then, find the sum of first (m-n) terms in terms of m and n, assuming m>n. Homework Equations Sum of an AP: n/2 * {2a+ (n-1)d} The Attempt at a Solution We get two equations: m= n/2 * {2a+...

The Beer-Lambert law gives the intensity of monochromatic light as a function of depth ##z## in the form of an exponential attenuation: $$I(z)=I_{0}e^{-\gamma z},$$ where ##\gamma## is the wavelength-dependent attenuation coefficient. However, if two different wavelengths are present...

Homework Statement I'm making a program to display what I have posted in the image and my program needs to look just like it. The idea is to have an employee enter information such as their pay and hours worked. Then taxes get calculated and it shows a net amount that the employee has made for...

Sir Michael Atiyah just gave a livestreamed talk claiming to prove the Riemann hypothesis. But it turns out that this is part of a larger research program in which he also claims to have an apriori calculation of the fine-structure constant and possibly other physical constants. Atiyah is 89...

Homework Statement Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence. Homework Equations The Attempt at a Solution I don't really know how to approach this problem, what the solution would...

Homework Statement The angles in triangle ABC form an increasing arithmetic sequence. The ratio of angles A:B:C can be written in the form 185:370:555 respectively. You are told that the total area of the triangle is 9 Length BC is Given the area of...

Homework Statement Homework Equations I'm guessing trigonometric identities such as sin(a)cos(b) = 1/2(sin(a+b)+sin(a-b)) might be relevant. The Attempt at a Solution I've been thinking of some way to get an approximation of each harmonic by working with the Fourier series representation...

QUESTION: The angles in triangle ABC form an increasing arithmetic sequence. The ratio of angles A:B:C can be written in the form 185:370:555 respectively. You are told that the total area of the triangle is 9 Length BC is Given the area of triangle ABC, work out the...

Arithmetic Series? Given the arithmetic series 5+14+23+...(to 241 terms), how many terms in the series are divisible by 5? I need a good explanation and a good start.

Need help with this I have 6 beaker of oil and 12 breaker of water, the total amount combined are 66 litres . What is the capacity of each Breaker"?

Encountered difficulties in proving the attached image. Greatly appreciate for the help!

Is there any very basic arithmetic book (for dummies) written by respected mathematicians like for example Serge Lang, Gelfand and Allendoerfer? Looking for a book that really explains the key concepts and rules of arithmetic, in a pedagogical way, from scratch, without skipping any steps when...

So I have this problem I'm stuck on wrapping my head around a particular problem "In the sequence a{n}, let a{0}=2. If a{n+1} = 3 a{n} −1, then what is the value of a3?" I understand it's following the pattern of each term, and that with Arithmetic sequence a{n-1} means you would use the a{n}...

A long time ago I read an explanation Richard Feynman did on how the concepts of arithmetic can be derived from basic principles, along the lines of Peano's axioms, but I don't remember where it was. Thanks.

Given two positive numbers a and b, we define the root mean square as follows: R. M. S. = sqrt{(a^2 + b^2)/2} The arithmetic mean is given by (a + b)/2. Given a = 1 and b = 2, which is larger, A. M. or R. M. S. ? A. M. = sqrt{1•2} A. M. = sqrt{2} R. M. S. = sqrt{(1^2 + 2^2)/2} R. M. S. =...

Given two positive numbers a and b, we define the geometric mean and the arithmetic mean as follows G. M. = sqrt{ab} A. M. = (a + b)/2 If a = 1 and b = 2, which is larger, G. M. or A. M. ? G. M. = sqrt{1•2} G. M. = sqrt{2} A. M. = (1 + 2)/2 A. M = 3/2 Conclusion: G. M. > A. M. Correct...

Problem Statement: Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact. Let μ be the first uncountable ordinal. The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former...

Problem Statement: Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact. Let μ be the first uncountable ordinal. The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former...

Hello everyone! So I was looking at Shor Algorithm for prime factorization and I have some doubts in the arithmetic part. Let's define a function f that : f(x) = ax mod N. The middle step in shor algorithm is to calculate, simultaneously, all values of f. In some papers and books, I saw some...

Question: Find the first 5 terms of this series and determine if it is an arithmetic sequence. An= 2 + 6n Help please!

Hi. Let's say I have data which I have measured. For example I measured a length of an object and the measurment was repeated 5 times. An instrument which I used to measure has an error, value of which I know. My options are to either to just go with the instrument error (probably not, right?)...

Hi everyone! I'm having trouble solving this problem, its set up in a way that I don't understand and I was hoping someone could help clarify it with me..Thanks a bunch!

What it does is described well in my comments. I'm using decimal, which is more precise than double but has a smaller range of values. https://msdn.microsoft.com/en-us/library/ms228360(v=vs.90).aspx const decimal pi =...

Say I have a vector product |x+a⟩⟨x| and I multiplied it by a ket vector |x'⟩. Can I pull the |x'⟩ into the ket vector |x+a⟩? also could you split up the ket vector |x+a⟩ into two ket vectors added together?

I have authored documents of 40 years of computer software development with a mind to collect them into a publication at some point. They have been built around several software topics but mathemetics is a favorite of mine. I find a point of inspiration and write a piece of software around it...

Supposing $x = 0$, do I need limits to solve $\frac{lnx }{x^2} + \frac{1}{2x^2}$? Since $lnx$ does not exist at $x = 0$, then the best we can do is $\lim_{{x}\to{0}} lnx$ which is $0$. But then, $x^2$ at zero equals 0, so we have $0/0$ which is indeterminate, so I need to find...

a1, a2, a3 and 4 make an arithmetic progression with difference d. For which values of d, A = a1a2 + a2a3 + a3a1 has the lowest value?I don't know if I went with the right approach, but I managed to get this : A=3x2 +6xd + 2d2 for a1= x, a2 = x + d, etc... But I don't know what else to do.

Homework Statement A grocer sells items by mixing small stones into grains. He has 50 kg each of two varieties of rice. He mixes 7.5 kg of stones in all. For one kg of rice, he mixes 100 grams of stones in the first variety and the remaining in the other. Peter purchases 5 kg of rice of each...

A competitor participating in a programme organised by a certain telelvision channel, wins by answering 15 questions correctly. The price money of 50 for the first question, 75 for second question , 100 for third question etc ... given for correct answers , are in an arithmetic progressionA...

I am new to number theory and I heard from my friend that we can use modular arithmetic to conveniently find the unit digit of a number or the remainder obtained on dividing a number by another number such as the remainder obtained on dividing (x^y) by a. Is it possible?How can we do this?

What happens when you have to subtract: 30 -9 Basically 30 - 9 but the 3 must be made 13, so what happens to the zero? I know if it was any other number we would reduce its value by one. Also, if possible, can anyone please show me how 30 in base 5 is 110? I understand that its 1*5*5 + 1*5...

I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ... I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...

I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ...I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...The...

Okay, So far I have seen and know to find the common difference of arithmetic terms using the consecutive terms. But this progression looks different (Shake) Find the common difference , Value of Z and the first term of it. Many Thanks (Happy)

Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...

Greetings. I've come with yet another issue which is causing me some distress. So I took everyone's advice and started self-teaching calculus (or at least I started trying... Limits are still confusing). However, my problem doesn't stem there. I decided to review my algebra before year 10...

I've encountered what seems to be two different notations for modular arithmetic and I'm confused as to whether they mean the same thing. My abstract algebra textbook (Pinter) and professor would write, for example, 5 = 15mod(10), as though mod(10) is an operation that returns the amount by...

I often read (for example, in Wikipedia on "Rosser's Trick") that in order for a proof of Gödel's First Incompleteness Theorem, one assumes an efficient consistent theory of numbers which includes a "sufficient fragment of elementary arithmetic". What minimum would qualify? Is Robinson's Q a...

Homework Statement A 25 year old programme for building new houses began in Core Town in the year 1986 and finished in 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in 2000 and 108 in 2010, find the number of houses built in 1986...

Here is a question that I have a problem with, It doesn't seem to have a solution: An increasing sequence that is made of 4 positive numbers, The first three of it are arithmetic series. and the last three are geometric series. The last number minus the first number is equal to 30. Find the sum...

Homework Statement A finite arithmetic progression is given such that ##S_n>0## and ##d>0##. If the first member of the progression remains the same but ##d## increases by 2, then ##S_n## increases 3 times. If the first member of the progression remains the same but ##d## increases 4 times...

Find the sum of the first 17 terms of the arithmetic series $$8+\sqrt{7}, \ 6,\ 4-\sqrt{7}$$ $$u=8+\sqrt{7}$$ $$S_{17} =\frac{u\left(1-\frac{{6}^{17}} {u} \right)}{u}$$ My first shot at this

*I am struggling with arithmetic and geometric sequences. if the 4th term is m-8, 6th term 8m+3 and 8th term is 10m-5 Calculate the 1st and 5th term Which term will have a value of -70The 4th term of geometric sequence is -16 and the 6th term is -64. Calculate the 3rd and 5th terms. thank you...

Homework Statement Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W. Homework Equations -Polynomial vector spaces -Subspaces...

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...

Homework Statement Let a, b, c be ordinals. Prove that a+(b+c)=(a+b)+c Homework EquationsThe Attempt at a Solution I looked at a set theory book by Jech and he says Prove by induction on c. Should I look at the case where its true for c+1[/B]

I am currently in upper division math courses in my career and I can't do something like 109-64 quickly in my head. I obviously try to break every number apart to make hard subtractions, but I can't do it fast!. Should I worry about this?. Overall, I have done well in my math life, but I feel...

Homework Statement What is the minimum number of digits to the right of the decimal point needed to express the fraction as a decimal? a) 4 b) 22 c) 26 d) 30 e) 104 Homework EquationsThe Attempt at a Solution One possible solution is: "We can rewrite the fraction as . Since the last digit of...