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},...
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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, ...
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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 * ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$
\...
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To find the sum of the series
It is a part of log function.
please help me to find the sum of this infinite series
$$1+e^{2x}+e^{6x}+e^{12x}+e^{20x}+\ldots$$
or
$$\sum_{n=1}^{\infty} e^{n(n-1) x}$$
I got this problem while ...
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What percentage of arithmic progressions cross the middle
Consider the first $n$ natural numbers and randomly pick $m$ numbers out of these. Define S as the set of these $m$ numbers, having density $d=m/n$. Consider all arithmic progressions of length 3 in S....
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Why are the $n$th-order differences of the sequences $a^n$ always equal to $n!$?
Take a look at this example with $a^2$ (where $a \in \mathbb{N}$).
...
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how to correctly compress expressions?
I don’t know mathematics well enough and I was faced with the task of rolling up such a thing. Could you tell me how to do this correctly?
$$ \frac{3}{x^3 + 1} + \frac{5}{x^3 + 1} + ... + \frac{2x + 1}...
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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 \...
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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 ...
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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
$$a+rq,\; a+(r+1)q,\; \...
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There are infinitely many non-harshad Fibonacci numbers
A harshad number is an integer that is divisible by the sum of its digits. For example, $280$ is a harshad number as it is divisible by $2+8+0=10$. Prove that there are infinitely many non-harshad ...
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Infinitely many primes of the form $\underbrace{11\dots 1}_{k \text{ times}}\dots \underbrace{11\dots 1}_{k \text{ times}}$
[Analytic Number Theory - Florian Luca and Jean Marie De Koninck, chapter 14, question 14.9]
Prove that for each positive integer $k$, there are infinitely many primes which when written in base $10$ ...
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When do disjoint sequences with bounded consecutive differences exist?
Let $N_1, N_2,\dots, N_n \in \mathbb{N}^{>1}$. On what conditions do disjoint increasing sequences $a_1, a_2, \dots, a_n: \mathbb{N} \rightarrow \mathbb{N}$ with consecutive differences bounded by $...
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On the sum of subsets of arithmetic progressions
Let it be an arithmetic progression $A = \{a_1, a_1+k, a_1+2k,...,a_1+(n-1)k : k\in\mathbb N\}$, and let $S=\{x_1,x_2,...,x_r\}$ be some subset of $A$, such that the elements of $S$ are equi-...
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Proof of conditions for polynomials
Find the conditions for the roots $\alpha, \beta, \gamma$ of the equation $x^3-ax^2+bx-c=0$ to be in: $(i)$A.P.; $(ii)$G.P.
If the roots are not in A.P. and if $\alpha+\lambda,\ \beta+\lambda,\ \...
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Why aren't $59$ L tiles the answer to this problem?
$$4*(1+3+5+7)=8^2.$$
$4*$(sum of odds from first to $k$th odd number)=area of a square region=$4*k^2$.
$k-1$ is the number of $L$'s.
You can solve the $10\times 12$ problem via considering $10\times ...
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What is $(3 n+1)+(3 n+2)+\ldots+(5 n)+(5 n+1)=$ equal to?
I'm trying to prove this inequality$\frac{1}{2}<\frac{1}{3 n+1}+\frac{1}{3 n+2}+\ldots+\frac{1}{5 n}+\frac{1}{5 n+1}<\frac{2}{3}$, and i came across some difficulties with arithmetic progression
...
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How fast does the proportion guaranteed by dirichlet converge?
I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
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Sieving a sequence of integers using an AP
Consider the sequence $1, 2, 3, \cdots, N$ where $N = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{ e_k}$ is the unknown factorization of a given $N$.
Define the set
$$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, ...
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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 ...
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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}...
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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 ...
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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
...
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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 ...
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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 ...
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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;
$$uN\{0,1,2,\...
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4th term of arithmetic progression is the sum of squares of the first three terms.
Given that $a,b,c,d \in Z, a\neq b\neq c \neq d$ are in arithmetic progression, and
$$d = a^2 + b^2 + c^2$$
Find the general term of the Arithmetic Progression.
My work:
Select a = $\alpha - \beta$, ...
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The choice of three or more consecutive terms of an A.P.
The story begins like..
I was reading through some pages of my highschool textbook....when I found this block of text
I should state that it is absolutely clear to me that why we consider three ...
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Does there exist a maximal subset of arithmetic progressions of length $k,$ such that any additional numbers will result in an A.P. of length $k+2?$
Fix $k\in\mathbb{N}.$ Definition: A set $A\subset\mathbb{N}$ (with $\vert A \vert = \infty$) is maximal of length $k$ if it contains at least one arithmetic progression of length $k,$ and, if $x\not\...
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What's the idea of Dirichlet’s Theorem on Arithmetic Progressions proof?
Dirichlet’s Theorem on Arithmetic Progressions says that if $a, m$ are natural numbers such that $gcd (a,m) = 1$, then there are infinitely many prime numbers in the arithmetic progression $a + km, k \...
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Proof for $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ can never be the terms of a single arithmetic progression with non zero common difference.
Proof for $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ can never be the terms of a single arithmetic progression with non zero common difference.
Here is what I have tried.
Let:
$$ax +c=\sqrt{2}$$
$$bx +c=\...
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3
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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 ...
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1
answer
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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 $\mathbb{Z}$ ...
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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 ...
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0
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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 ...
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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 ...
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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 ...
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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}-\...
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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 $\alpha$, $\alpha+d$, $\alpha+2d$, $\alpha+3d$, $\alpha+4d$ ...
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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 ...
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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 ...
