Questions tagged [arithmetic-progressions]

Questions related to arithmetic progressions, which are sequences of numbers such that the difference between consecutive terms is constant

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Is the sequence $\underbrace{2,2,...2}_{1000\text{times}}$ a harmonic sequence?

Is the sequence: $\underbrace{2,2,...2}_{1000\text{times}}$ a harmonic sequence? First of all let me give a bit introduction about what is a harmonic sequence. For e.g.: We can say that $1,\frac{1}{2},...
Syamaprasad Chakrabarti's user avatar
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If Erdős Conjecture on arithmetic progressions is true, and $A$ is large, then does there exist a consecutive A.P. of $A$ of length $k$ for every $k?$

Question 1: If Erdős Conjecture on arithmetic progressions is true, and $A\subset \mathbb{N}$ is large, then does $A$ contain a consecutive A.P. of $A$ of length $k;\ \lbrace{a_{m+1}, a_{m+2},\ldots, ...
Adam Rubinson's user avatar
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Logical progression of division?

A while back, I was introduced to the concept of tetration with a preface of going from simple addition (a + 1 b number of times) through multiplication (a + a b number of times), exponentiation (a * ...
Vedrit's user avatar
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What is the least number of terms $a+nd$ required for a finite arithmetic progression?

I would like clarification on the following definition of finite arithmetic progression: According to Wikipedia, "A finite portion of an arithmetic progression is called a finite arithmetic ...
M. B. Jones's user avatar
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Regarding ratio of sum of arithmetic sequence.

Consider two arithmetic progressions, $\langle a_n\rangle_{n \in N}$ and $\langle b_n\rangle_{n \in N}$, such that $$\frac{\sum_{r=1}^n a_r}{\sum_{r=1}^n b_r} = \frac{3n+1}{4n+2}$$ Find the ratio of ...
Sahaj's user avatar
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Incorrect partial sum formula in textbook?

I was helping my brother with his maths homework, where he has just started learning about arithmetic series and their formulas such as the sum of the first $n$ terms ($S_n$) or finding the $n$th term ...
cherrytree's user avatar
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N-digit geometric numbers which relate to arithmetic progression

Call a $3$-digit number geometric if it has $3$ distinct digits which, when read from left to right, form a geometric sequence. So, consider the number $931$. Let us note $931-792=139$ which means ...
Mikhail Gaichenkov's user avatar
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Could you prove van der Waerden's theorem with a probabilistic argument

If we would try to prove the simplest case, that is: if we color the integers with 2 colors, then the coloring must contain an arithmetic progression of length 3. Let $R_n$ be a randomly generated ...
AndroidBeginner's user avatar
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Intersection of natural sets [duplicate]

I'm triying to compute $$ (a \mathbb{N} + b) \cap (c \mathbb{N} + d). $$ I've tried to solve $$ ax + b = cy + d $$ I don't get a exact result. I think that the intersecction is something like $$ \...
JulianDoyle's user avatar
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To find the sum of the series

It is a part of log function. please help me to find the sum of this infinite series $$1+e^{2x}+e^{6x}+e^{12x}+e^{20x}+\ldots$$ or $$\sum_{n=1}^{\infty} e^{n(n-1) x}$$ I got this problem while ...
Arun Kotagi's user avatar
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What percentage of arithmic progressions cross the middle

Consider the first $n$ natural numbers and randomly pick $m$ numbers out of these. Define S as the set of these $m$ numbers, having density $d=m/n$. Consider all arithmic progressions of length 3 in S....
AndroidBeginner's user avatar
2 votes
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Why are the $n$th-order differences of the sequences $a^n$ always equal to $n!$?

Take a look at this example with $a^2$ (where $a \in \mathbb{N}$). ...
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how to correctly compress expressions?

I don’t know mathematics well enough and I was faced with the task of rolling up such a thing. Could you tell me how to do this correctly? $$ \frac{3}{x^3 + 1} + \frac{5}{x^3 + 1} + ... + \frac{2x + 1}...
ZiEnTenIn's user avatar
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Doubt in simple JEE Arithmetic Progression question

Question from JEE Question bank: If $a_1 = 50$ and $a_1 + a_2 + a_3 + .... + a_n = n^{2}a_n \;\;\forall \;n \geq 1$ then $a_{100}$ equals: a)$1/100$ b)$1/101$ c)$1/50$ d)$1/51$ Here, the correct ...
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Prove that $a_n=a_{n-1}$ [duplicate]

Suppose the set $\mathbb{Z}_{\ge 0}$ of nonnegative integers is partitioned into finitely many arithmetic progressions of the form $a_i \mathbb{Z}_{\ge 0} + b_i$ with $1\leq i\leq n, b_i\ge 0$ and $1\...
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Find proportion $k$ between two arithmetic series such that $S_1(i) \ge S_2(i) k$

Let's say that: $S_1(i) = a + (i - 1) b$ $S_2(i) = c + (i - 1) d$ and I want to find the k that fulfils $S_1(i) \ge S_2(i) k$ I've tried doing the inequation $a + (i - 1) b \ge (c + (i - 1) d ) k \...
cdecompilador's user avatar
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Can an imaginary sequence be produced to ease the equation of general term of a series whose common difference is in an AP

I came across a problem which follows the series: $2,3,6,11,18$ and so on.. It can be observed that the common difference of the series was in an arithmetic progression.. Using the regular method of ...
umit-adlakha's user avatar
2 votes
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If $0<q<2000$, then at most $10$ consecutive terms of the arithmetic progression $a,\; a+q,\; a+2q\dots$ can be primes

Show that if $0<q<2000$, then at most $10$ consecutive terms of the arithmetic progression $$a,\; a+q,\; a+2q\dots$$ can be primes. If not, then there is an $r$ such that $$a+rq,\; a+(r+1)q,\; \...
Sayan Dutta's user avatar
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6 votes
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There are infinitely many non-harshad Fibonacci numbers

A harshad number is an integer that is divisible by the sum of its digits. For example, $280$ is a harshad number as it is divisible by $2+8+0=10$. Prove that there are infinitely many non-harshad ...
Sayan Dutta's user avatar
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Infinitely many primes of the form $\underbrace{11\dots 1}_{k \text{ times}}\dots \underbrace{11\dots 1}_{k \text{ times}}$

[Analytic Number Theory - Florian Luca and Jean Marie De Koninck, chapter 14, question 14.9] Prove that for each positive integer $k$, there are infinitely many primes which when written in base $10$ ...
Sayan Dutta's user avatar
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When do disjoint sequences with bounded consecutive differences exist?

Let $N_1, N_2,\dots, N_n \in \mathbb{N}^{>1}$. On what conditions do disjoint increasing sequences $a_1, a_2, \dots, a_n: \mathbb{N} \rightarrow \mathbb{N}$ with consecutive differences bounded by $...
user242318's user avatar
2 votes
1 answer
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On the sum of subsets of arithmetic progressions

Let it be an arithmetic progression $A = \{a_1, a_1+k, a_1+2k,...,a_1+(n-1)k : k\in\mathbb N\}$, and let $S=\{x_1,x_2,...,x_r\}$ be some subset of $A$, such that the elements of $S$ are equi-...
Juan Moreno's user avatar
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1 vote
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Proof of conditions for polynomials

Find the conditions for the roots $\alpha, \beta, \gamma$ of the equation $x^3-ax^2+bx-c=0$ to be in: $(i)$A.P.; $(ii)$G.P. If the roots are not in A.P. and if $\alpha+\lambda,\ \beta+\lambda,\ \...
J_dash's user avatar
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Why aren't $59$ L tiles the answer to this problem?

$$4*(1+3+5+7)=8^2.$$ $4*$(sum of odds from first to $k$th odd number)=area of a square region=$4*k^2$. $k-1$ is the number of $L$'s. You can solve the $10\times 12$ problem via considering $10\times ...
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What is $(3 n+1)+(3 n+2)+\ldots+(5 n)+(5 n+1)=$ equal to?

I'm trying to prove this inequality$\frac{1}{2}<\frac{1}{3 n+1}+\frac{1}{3 n+2}+\ldots+\frac{1}{5 n}+\frac{1}{5 n+1}<\frac{2}{3}$, and i came across some difficulties with arithmetic progression ...
SysRq308's user avatar
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How fast does the proportion guaranteed by dirichlet converge?

I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
Bruno Andrades's user avatar
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Sieving a sequence of integers using an AP

Consider the sequence $1, 2, 3, \cdots, N$ where $N = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{ e_k}$ is the unknown factorization of a given $N$. Define the set $$M = \left\{\overbrace{p_1, 2p_1, \cdots,...
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Arithmetic progression and proof by induction/contradiction

Show that no cube of an integer can be expressed as $7n + 5$ for some positive integer $n$ This is from Riley's "Mathematical methods for Physics and Engineering", and is question 1.28 b, ...
Dimbles's user avatar
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Finding or parametrizing integer solutions to $pq(p^2-q^2)=rs(r^2-s^2)$

Background: The order-3 magic square of squares problem (MSS3) is a well-known open problem that involves finding eight separate arithmetic progressions of three squares (APSs). In particular, two ...
Eric Snyder's user avatar
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Summation of an Arithmetico-Geometric Progression

$$ \sum_{r=1}^{11} r5^r = \frac{43\times5^{a}+ 5}{b}$$ Find (a+b) This question was asked in an exam. I got the answer 28. However the answer given was 15. Here is my attempt: Let S = $ \sum_{r=1}^{11}...
BlackHood's user avatar
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Has this weak version of Erdős Conjecture on arithmetic progressions been proven, or is it still an open problem?

This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker version of Erdős Conjecture, but I do not know how to prove it. Erdős conjecture on arithmetic progressions ...
Adam Rubinson's user avatar
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3 answers
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If $\vert A + A \vert = 2\vert A \vert - 1,$ then is $A$ an arithmetic progression?

For $n\in\mathbb{N},$ define $[n]:= \{ 0, 1, 2,\ldots, n-1\}.$ Suppose $A$ is a finite subset of $\mathbb{Z}$ such that $\vert A + A \vert = 2\vert A \vert - 1,$ where $\ A+B$ means the Minkowski ...
Adam Rubinson's user avatar
3 votes
2 answers
160 views

The correct solution for a 10th grade A.P. problem

Show that $a_1$, $a_2$,...,$a_n$,... form an AP where $a_n$ is defined as below: $a_n = 3+4n$ This is a problem from a grade 10th textbook... I solved the question in a different way from how our ...
GameTime With Aryan's user avatar
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Square terms in an AP

Suppose we have an Arithmetic Progression given by $$ a_n = a_0 + nd $$ Suppose $a_0 = k^2$ is a perfect square. We are looking for $m$ such that $a_m = s^2$. Is there an intelligent way to search for ...
vvg's user avatar
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1 vote
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Prove that $uN\{0,1,2,\dots,m-1\}\equiv N\{0,1,2,\dots,m-1\}\pmod{mN}$, when gcd$(mN,u)=1$

Let $m,N\in\mathbb{N}_2$, and consider modulo $mN$. Let $u\in\{1,\dots,mN-1\}$ such that gcd$(mN,u)=1$. I want to show that the following two sets are equal, where order is unimportant; $$uN\{0,1,2,\...
MeBadMaths's user avatar
1 vote
1 answer
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4th term of arithmetic progression is the sum of squares of the first three terms.

Given that $a,b,c,d \in Z, a\neq b\neq c \neq d$ are in arithmetic progression, and $$d = a^2 + b^2 + c^2$$ Find the general term of the Arithmetic Progression. My work: Select a = $\alpha - \beta$, ...
Sahaj's user avatar
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The choice of three or more consecutive terms of an A.P.

The story begins like.. I was reading through some pages of my highschool textbook....when I found this block of text I should state that it is absolutely clear to me that why we consider three ...
Darshit Sharma's user avatar
2 votes
0 answers
64 views

Does there exist a maximal subset of arithmetic progressions of length $k,$ such that any additional numbers will result in an A.P. of length $k+2?$

Fix $k\in\mathbb{N}.$ Definition: A set $A\subset\mathbb{N}$ (with $\vert A \vert = \infty$) is maximal of length $k$ if it contains at least one arithmetic progression of length $k,$ and, if $x\not\...
Adam Rubinson's user avatar
7 votes
1 answer
508 views

What's the idea of Dirichlet’s Theorem on Arithmetic Progressions proof?

Dirichlet’s Theorem on Arithmetic Progressions says that if $a, m$ are natural numbers such that $gcd (a,m) = 1$, then there are infinitely many prime numbers in the arithmetic progression $a + km, k \...
Nicolás A.'s user avatar
3 votes
1 answer
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Proof for $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ can never be the terms of a single arithmetic progression with non zero common difference.

Proof for $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ can never be the terms of a single arithmetic progression with non zero common difference. Here is what I have tried. Let: $$ax +c=\sqrt{2}$$ $$bx +c=\...
Arjun Agrawal's user avatar
2 votes
3 answers
351 views

Solve the equation $x^4-2x^3-21x^2+22x+40=0$ whose roots are in A.P.

Solve the equation $x^4-2x^3-21x^2+22x+40=0$ whose roots are in A.P. (arithmetic progression). I don't understand this solution. Why are the terms of AP considered as mentioned in the question and ...
B 10 Dhuri Jayant Gangadhar's user avatar
4 votes
1 answer
124 views

Coloring arithmetic progression

I've been looking at some old notes from the course Probabilistic Combinatorics and I saw the following question: Prove that there exists a constant $N$ and a red/blue coloring of $\mathbb{Z}$ ...
DIexp's user avatar
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An elementary proof that there are no 4 squares in arithmetic progression

My problem. I am trying to fill in the details in @lhf 's outline of an elementary proof (mentioned in Dickson's History of the theory of numbers) that there are no four Squares in arithmetic ...
UniformConvergence's user avatar
1 vote
0 answers
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Clarification of the key theorem for proving Roth's theorem

I have recently been reading Terence Tao's paper on the Szemeredi's original proof of Szemeredi's theorem (https://terrytao.files.wordpress.com/2017/09/szemeredi-proof1.pdf) and I have come upon this ...
sgvozdic's user avatar
0 votes
2 answers
142 views

Finding the sum of the first 10 terms within an Arithmetic Progression

I have been attempting to solve the following question, however, I am unable to form any sort of relationship between the two facts. I have attempted at using all three equations, however am unable to ...
Ali's user avatar
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3 answers
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What's the value of $p$ if the roots of the biquadratic equation $x^4-10x^2+p=0$ are in AP?

What's the value of $p$ if the roots of the biquadratic equation $$x^4-10x^2+p=0$$ are in AP? The given equation is quadratic in $x^2$, so it's discriminant is $D=25-p\ge0\iff p\le25$ and the roots ...
yinivem462's user avatar
1 vote
2 answers
119 views

The cubic $x^3+ax^2+bx+c$ has three distinct zeros in GP and the reciprocal of these zeros are in AP then prove that $2b^2+3ac=0$

I tried to solve this question: first assuming the zeros to be $m$, $mr$ and $mr^2$ in G.P. so that $\frac{1}{m}$, $\frac{1}{mr}$ and $\frac{1}{mr^2}$ in A.P.; then by solving it like $\frac{1}{mr}-\...
Anubhav Panchal's user avatar
0 votes
4 answers
182 views

Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$, given that the roots form an arithmetic progression.

Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$, given that the roots form an arithmetic progression. What I've tried : Letting $\alpha$, $\alpha+d$, $\alpha+2d$, $\alpha+3d$, $\alpha+4d$ ...
Ash_Blanc's user avatar
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Arithmetic progressions in arrays with limited range

It is known that finding 3 evenly spaced ones and 3SUM applied to (sorted) arrays with elements in the range $[0, \dots, N]$ can be solved in $\mathcal{O}(n + N \log N)$ time. Methods for solving ...
jorisperrenet's user avatar
0 votes
3 answers
40 views

mathematical expression for arithmetic sequence with 3 variables where 1 of them is known

This is from an A-level exam paper, so it's fairly elementary. I'd like the know how to express the following mathematically. A construction programme began in 1986 and finished in 2010. The number ...
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