# 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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### 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 ...
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### 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 ; $1,2,4,8,...,$ ...
• 129
1 vote
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### 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 ...
• 35
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### 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 ...
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### 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 ...
• 1,190
1 vote
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### 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 ...
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 ...
• 311
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### 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 ...
• 9,552
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### 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....
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• 101
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### 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 ...
• 17
1 vote
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### 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 ...
• 3,835
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### 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 $n \neq m$. ...
• 243
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### 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 ...
• 491
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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 $k;\ \lbrace{a_{m+1}, a_{m+2},\ldots, ... • 20.7k 1 vote 1 answer 52 views ### What is the least number of terms$a+nd$required for a finite arithmetic progression? I would like clarification on the following definition of finite arithmetic progression: According to Wikipedia, "A finite portion of an arithmetic progression is called a finite arithmetic ... 2 votes 0 answers 44 views ### Regarding ratio of sum of arithmetic sequence. Consider two arithmetic progressions,$\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 ... • 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 ... • 553 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 ... 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 ... 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 ... 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.... 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$a \in \mathbb{N}$). ... • 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? $$\frac{3}{x^3 + 1} + \frac{5}{x^3 + 1} + ... + \frac{2x + 1}... 1 vote 0 answers 112 views ### Doubt in simple JEE Arithmetic Progression question Question from JEE Question bank: If a_1 = 50 and a_1 + a_2 + a_3 + .... + a_n = n^{2}a_n \;\;\forall \;n \geq 1 then a_{100} equals: a)1/100 b)1/101 c)1/50 d)1/51 Here, the correct ... • 331 1 vote 0 answers 53 views ### Prove that a_n=a_{n-1} [duplicate] Suppose the set \mathbb{Z}_{\ge 0} of nonnegative integers is partitioned into finitely many arithmetic progressions of the form a_i \mathbb{Z}_{\ge 0} + b_i with 1\leq i\leq n, b_i\ge 0 and 1\... • 869 0 votes 0 answers 18 views ### Find proportion k between two arithmetic series such that S_1(i) \ge S_2(i) k Let's say that: S_1(i) = a + (i - 1) b S_2(i) = c + (i - 1) d and I want to find the k that fulfils S_1(i) \ge S_2(i) k I've tried doing the inequation a + (i - 1) b \ge (c + (i - 1) d ) k \... 0 votes 0 answers 33 views ### Can an imaginary sequence be produced to ease the equation of general term of a series whose common difference is in an AP I came across a problem which follows the series: 2,3,6,11,18 and so on.. It can be observed that the common difference of the series was in an arithmetic progression.. Using the regular method of ... 2 votes 3 answers 152 views ### If 0<q<2000, then at most 10 consecutive terms of the arithmetic progression a,\; a+q,\; a+2q\dots can be primes Show that if 0<q<2000, then at most 10 consecutive terms of the arithmetic progression$$a,\; a+q,\; a+2q\dots$$can be primes. If not, then there is an r such that$$a+rq,\; a+(r+1)q,\; \... • 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 ... • 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$10$... • 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$...
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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-...