Skip to main content

Questions tagged [arithmetic-progressions]

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

Filter by
Sorted by
Tagged with
9 votes
1 answer
292 views

Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$

Let $X:=\{$ positive integers that contain the digit $2\}$ For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$ I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap ...
Adam Rubinson's user avatar
3 votes
2 answers
234 views

Interesting NT Question With AP and GCD.

Find the number of prime triplets $(p, q,r)$ such that $p(p + 1), q(q + 1),r(r + 1)$ form a strictly increasing arithmetic progression, where GCD $(r − p, 2p + 1)=1$. What I tried: $r(r+1)-p(p+1)=2d$ ...
CLASH ROYAL's user avatar
0 votes
1 answer
31 views

Quadratic where roots and coefficients together form Arithmetic Progression

Background I was reading this post: A.P. terms in a Quadratic equation. And wondered the following: Given a quadratic $ax^2+bx+c=0$ which has roots $x=m,x=n$, is it possible for $a,m,b,n,c$ to be ...
Red Five's user avatar
  • 2,782
1 vote
1 answer
63 views

How to form an AP which contains common terms of two other APs?

I got this question when I was going through a question from the infamous entrance exam JEE. It was a question from the JEE Advanced 2018, Paper 1. The question is as follows: Let X be the set ...
Parithiilamaaran.H's user avatar
2 votes
2 answers
71 views

Given $a,b,c,d$ in arithmetic progression, can we express the solution to $\frac1{x-a}+\frac1{x-b}+\frac1{x-c}+\frac1{x-d}=0$ in terms of $a,d$ only?

Context This is a question I wrote for myself recently, heavily based on an old (1930s) examination paper for university admissions. Given $a,b,c,d$ which are real numbers and consecutive terms in an ...
Red Five's user avatar
  • 2,782
2 votes
1 answer
56 views

Exercise $3.2.6$ in Tao-Vu's Book - why is it trivial for $\epsilon \geqslant 8^{-d}$?

Let $P$ be a proper progression of rank $d$ in an additive group $(Z,+)$, and let $A\subset P$ such that $|A| \leqslant ε|P|$ for some $0 <ε< 1.$ Show that $P\setminus A$ contains a proper ...
stoic-santiago's user avatar
0 votes
1 answer
29 views

Arithmetic Progressions involving 3 distinct terms

I've been trying to find the number of Arithmetic Progressions involving 3 distinct terms of the set: $ \sum_{i=0}^{2021} a_i \cdot 3^i, \quad \text{where } a_i \in \{0, 2\}$ But I've had no sucess, ...
João Víctor Melo's user avatar
1 vote
1 answer
32 views

Application of $a_n$ = $S_n$ - $S_{n-1}$ for an arithmetic progression.

Consider the following question: Let $V_r$ denote the sum of the first r terms of an Arithmetic Progression whose first term is r and common difference is (2r-1). Let $T_r$ = $V_{r+1}$ - $V_r$ -2. ...
Bread Butter's user avatar
0 votes
1 answer
61 views

$\frac{a}{x},\frac{b}{y},\frac{c}{z}$ are in HP

If non-zero numbers $a,b,c,x,y$ and $z$ are such that $a,b,c$ are in AP, $x,y,z$ are in GP and $\frac{a}{x},\frac{b}{y},\frac{c}{z}$ are in HP then prove $|a|=|c|$. I have been trying to solve this ...
Vikas Sharma's user avatar
3 votes
2 answers
58 views

Prove $p,q,r$ are AP if $p\left(\frac{1}{q}+ \frac{1}{r}\right), q\left(\frac{1}{r}+ \frac{1}{p}\right), r\left(\frac{1}{p}+ \frac{1}{q}\right)$ is AP

How should one approach this question. The following is my attempt: I started by forming the following equation, which is the inherent property of Arithmetic Progression: $$2 \cdot q\left(\frac{1}{r}+ ...
Epimu Salon's user avatar
6 votes
4 answers
1k views

Relationship between the squares of first n natural numbers and first n natural odd numbers.

Here's a question from high school mathematics. If $ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + \dots + 100^2 = x $, then ($1^2 + 3^2 + 5^2 + \dots + 99^2$) is equal to ? Options were: (a) $\frac{x}{2}-2525$ (b) ...
Ishant's user avatar
  • 63
0 votes
2 answers
52 views

I have got stucked with this concept of A.P.

Q) How to prove that the sequence:$ 2,4,6,8,...,1000$ is an A.P.$($$Arithmetic$ $Progression$)? First of all, the $1^{st}$ term of this sequence is $2$ and the common difference of this sequence is ...
Dropper's user avatar
  • 129
0 votes
0 answers
52 views

Formula for calculating the sum of the equation: $y = \lfloor 400(x-6)^{1.1} \rfloor$

I have an equation of $y = \lfloor 400(x-6)^{1.1} \rfloor$ where x is equal to or greater than 6 and increases by an increment of 1. I want to calculate what the sum of the equations added up together ...
xiao xiao's user avatar
-2 votes
2 answers
84 views

What is the formula for finding the summation of the sequence : $1,2,5,12,26,51,...$ upto $n$ terms? [duplicate]

Q)What is the formula for finding the summation of the sequence $1,2,5,12,26,51,...$ upto $n$ terms ? I know how to find the summation of sequences like $1,2,3,...,$ upto $n$ terms ; $1,2,4,8,...,$ ...
Dropper's user avatar
  • 129
1 vote
0 answers
105 views

Can any of you do something relevant with this mathematical property I found? [closed]

I'm an amateur mathematician and I found a property that I've never seen anyone mention before, I think I managed to demonstrate it below. I confess that I don't know if it's new or not, as I haven't ...
Edu's user avatar
  • 35
0 votes
3 answers
88 views

General term for increasing AP's

Can some give an easy general method to find general term of sequences whose difference is in AP? Example: 1,4,8,13,19.... The difference is 3,4,5... which is in AP. Through vigorous testing and ...
Maths lover's user avatar
0 votes
0 answers
52 views

Conjecture on digits in arithmetic progression

Let $n\geq 100$ be some positive integer in base $10$. Take its consecutive digits in groups of $k$ digits each. I have the following Conjecture: If the digits of some positive integer $n\geq 100$ in ...
Juan Moreno's user avatar
  • 1,190
1 vote
0 answers
30 views

If for any element's neighbors' average equals to element in sequence, it is an arithmetic progression

I need to prove that if a sequence $\{a_n\}_{n\in\Bbb N}$ is such that $$ a_n = \frac{a_{n-1}+a_{n+1}}{2} \quad\forall n\in\Bbb N $$ then the sequence is arithmetic progression. I transformed that ...
James Sup's user avatar
2 votes
1 answer
144 views

$\mathcal S =\{ a^{x_i} + a^{x_j} + a^{x_k} \mid i,j,k \in \{1,2,3, \ldots,n \} , i+j+k=n \}$

The statement of the problem : Let $n \in \mathbb Z$ , $n \ge 3$ and consider the real numbers $ 0 \le x_1 \lt x_2 \lt x_3 \lt \ldots \lt x_n$ in arithmetic progression and also consider the real ...
Last X's user avatar
  • 311
5 votes
1 answer
132 views

Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$

I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
Sayan Dutta's user avatar
  • 9,552
2 votes
1 answer
124 views

given that $a_n$ is an arithmetic sequence, and $a_{14}^2+a_{15}^2+a_{16}^2=35$, $a_{15} > 0$ find the general term of an $a_n$

I am right now in the Hebrew University Academic Prep School which in its level is equivalent to an American high school/college. It's a one-year academic prep school for engineering and exact science....
David's user avatar
  • 21
2 votes
1 answer
46 views

Question where sum of terms of an AP is a trignometric function

The question is as such If the sum of the first $n$ terms of an arithmetic progression is denoted by $S_n$ and $$S_n = 6n\sec^2 \theta + n(n-1)(\sin^2 \theta(4 + \tan^2 \theta) + \cos^2 \theta(4 + \...
koiboi's user avatar
  • 356
1 vote
2 answers
75 views

Shifted start arithmetic progression formula why it works? $a_n=a_k+(n-k)\cdot d$

Question:The number of zeros in $(10^{60}+1)^2$ is? The number of zeros in $(10^1+1)^2$ is zero. The number of zeros in $(10^2+1)^2$ is two. The number of zeros in $(10^3+1)^2$ is four. There's a ...
user avatar
4 votes
1 answer
86 views

counting number of arithmetic sequenses of diff 1 of length of at least 2 that could be made from numbers {1,2,...,n}

im asked to find the number of arithmetic sequences of d=1 (or consecutive sequences the way the question describes it), of length of at least 2 from the numbers {1,2,...,n}. the question adds that it ...
user1188938's user avatar
0 votes
1 answer
49 views

Arithmetic-Geometric progression general integral formula

I'm solving Arithmetic-Geometric progression. It's rules: $$ k(x)\in Z, \forall{x}\\ f(0)=k(0)\\ q(x) > 0, \forall{x}\\ f(x)=f(x-1)\frac{q(x)}{q(x-1)}+k(x)-k(x-1) $$ I got the general formula: $$ f(...
dgzargo's user avatar
  • 101
0 votes
1 answer
88 views

Why is $\sum_{k=1}^n {ka_k} = \frac{n(n+1)}{2}\frac{(a_1+2a_n)}{3}$ where $(a_n)$ is an arithmetic progression. [closed]

This is a question about the sum of an arithmetic sequence. Please forgive my lack of experience with LaTex. To find the sum as asked, I wrote a general term for the arithmetic sequence and used the ...
TH_Lee's user avatar
  • 17
1 vote
1 answer
115 views

Assuming all the terms of A.P. $[a_{n}]$ are integers with $a_{1}=2019$. Find number of such sequence satisfying given condition

Assuming all the terms of A.P. $[a_{n}]$ are integers with $a_{1}=2019$ and for $n\in I^+$ there always exsits positive integer $m$ such that $a_{1}+a_{2}+.....+a_{n}=a_{m}$, then find number of such ...
mathophile's user avatar
  • 3,835
3 votes
4 answers
165 views

Why does $n*T_n = m*T_m$ imply $T_\text{n+m} = 0$?

I was fiddling around with Arithmetic Progressions and I noticed this pattern. \begin{align} n*T_n = m*T_m \implies T_\text{n+m} = 0 \end{align} where $n, m \in \{0, \mathbb{Z}^{+}\}$ and $n \neq m$. ...
HerrAlvé's user avatar
  • 243
2 votes
0 answers
88 views

When is the sum of reciprocals of positive integers convergent?

I'm looking for sufficient conditions on an infinite $\Lambda\subseteq\mathbb{Z}_+$ so that $$\sum_{n\in\Lambda}\frac{1}{n}<\infty.$$ I know that the contraposition of this question is given by ...
Miles Gould's user avatar
1 vote
3 answers
111 views

if $a_1,a_2...a_n $ are in AP and the first 16 terms add up to 114 , find $a_1+a_6+a_{11}+a_{16}$

If $a_1,a_2...a_n $ are in an Arithmetic Progression (AP) such that the sum of the first $16$ terms is $114$, find $a_1+a_6+a_{11}+a_{16}$. My attempt: The sum of the first 16 terms is given by $8(...
math and physics forever's user avatar
0 votes
0 answers
84 views

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
0 votes
0 answers
36 views

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
1 vote
1 answer
52 views

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
2 votes
0 answers
44 views

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
  • 3,298
8 votes
1 answer
1k views

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
0 votes
0 answers
32 views

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
0 votes
0 answers
40 views

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
0 votes
1 answer
111 views

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
0 votes
1 answer
53 views

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
1 answer
222 views

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 $a \in \mathbb{N}$). ...
HerrAlvé's user avatar
  • 243
0 votes
0 answers
68 views

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
1 vote
0 answers
112 views

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 ...
Bongo Man's user avatar
  • 331
1 vote
0 answers
53 views

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\...
Alfred's user avatar
  • 869
0 votes
0 answers
18 views

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
0 votes
0 answers
33 views

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
3 answers
152 views

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
  • 9,552
6 votes
1 answer
129 views

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
  • 9,552
6 votes
1 answer
342 views

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
  • 9,552
6 votes
3 answers
233 views

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
87 views

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
  • 1,190

1
2 3 4 5
21