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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31 views

Given an arithmetic sequence a;b;c and a+b+c=abc, find a;b;c

At first glance, I immediately realize that a=b=c=0 is a solution, but I’m wondering if there are other solutions. Assuming other solutions exists, I manipulated the equation and got ac=3. But from ...
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1answer
44 views

prove :nth root of the Product of arithmetic sequence is bigger than the sqrt of the first and last numbers

let $ \{a_n\} $ be an arithmetic sequence of positive numbers. prove that for all $ n\ge2 $: $ \frac{a_{1}+a_{n}}{2}\ge\sqrt[n]{a_{1}a_{2}a_{3}\dots a_{n}}\ge\sqrt{a_{1}a_{n}} $ I managed to solve the ...
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1answer
54 views

Colouring of elements of set

Each of the numbers in the set $A = \{1, 2, ...., 2020\}$ is coloured either red or white. Prove that for $n \geq 18$, there exists a colouring of the numbers in A such that any of its $n$-term ...
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55 views

Is there a tight bound on following binomial summations involving squares on arithmetic progressions?

The summations of interest is following: $$\sum_{i=0}^{\lfloor\sqrt n\rfloor}\binom{n}{i^2}$$ $$\sum_{i\in\{a,q+a,2q+a\dots,\lfloor\sqrt n\rfloor\}}\binom{n}{i^2}$$ where $q<n$ and $a\in\{0,1,\dots,...
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How to get $A$ and $B$ from $A\csc 10^\circ+B=$ $\sin 10^\circ+\cos 60^\circ+\cos 40^\circ+\sin 70^\circ+\sin 90^\circ$?

The problem is as follows: Find $A+B$ from: $A\csc 10^\circ+B=\sin 10^\circ+\cos 60^\circ+\cos 40^\circ+\sin 70^\circ+\sin 90^\circ$ The alternatives given in my book are as follows: $\begin{array}{ll}...
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1answer
75 views

Inequality proof $\frac{n}{a_1a_{2n+1}}<\frac{1}{a_1a_2}+\frac{1}{a_3a_4}+…+\frac{1}{a_{2n-1}a_{2n}}<\frac{n}{a_0a_{2n}}$

Can someone give me an idea about how to prove the inequality? Mention that $a_0, a_1, a_2, ..., a_{2n}, a_{2n+1}$ is an AP and $0<a_0<a_1<...<a_{2n}<a_{2n+1}$ $$\frac{n}{a_1a_{2n+1}}&...
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2answers
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How to find the first term common to 2 AP [duplicate]

Suppose there are 2 APs, AP1 and AP2 where the first terms are a1 and a2 respectively and the common difference is d1 and d2. How do you find a mathematical formula to define the AP of the common ...
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3answers
105 views

$x^4-mx^3-2mx^2+2m^2x=0$ roots in arithmetic progression

For which real values of $m$ roots of equation $$x^4-mx^3-2mx^2+2m^2x=0$$ are in arithmetic progression? I managed to find a solution using a lot of casework, i.e. factoring the equation into $$x(x-m)(...
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Three-term arithmetic progression modulo $n$

Given five positive integers $a, b, c, d, n$, is it legitimate to call a triple $(a, b, c)$ an arithmetic progression modulo $n$, if $b = a + d \mod n$ and $c = b + d \mod n$?
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1answer
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Meaning of “non-trivial three-term arithmetic progressions”

In this paper "ON ROTH’S THEOREM ON PROGRESSIONS" by Tom Sanders, the author gives a bound related to "non-trivial three-term arithmetic progressions", but what exactly means "...
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1answer
76 views

If an arithmetic progression contains a perfect square, then it must contain a perfect square strictly less than …

(Question): If an arithmetic progression of positive integers $a, a+d, a+2d, \dots$ contains a perfect square, then it must contain a perfect square strictly less than $a+2d\sqrt{a}+d^2$. I noticed ...
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1answer
34 views

What are some good resources to learn arithmetic combinatorics?

I have been very intrigued by the theory of arithmetic combinatorics after the result regarding primes in arithmetic progressions proven by Tao and Green. PAPERS ARE OK. I notice that most people ...
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83 views

choosing large subset without arithmetic progression

I have stumbled upon this problem in a Number Theory book and got stuck with it. The problem states: Given the set $(1,2,...,N)$, Prove that if $N$ is large enough one can choose a subset of at least $...
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1answer
136 views

Interesting BMO Final Selection Test Problem 1998 Q3

An infinite arithmetic progression, whose terms are all positive integers, contains (i) a perfect square which is not a perfect cube, and (ii) a perfect cube which is not a perfect square. Prove that ...
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Which term of the A.P. $3, 15, 27, 39, …$ will be $132$ more than its 54th term? [closed]

Which term of the A.P. $3, 15, 27, 39, …$ will be $132$ more than its 54th term? Although this is a ncert question still I am asking because not able to solve it.
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22 views

The sides of a triangle are in A.P.

One of the angles of a triangle is $120^\circ$. The sides of the triangle are in A.P. Find the ratio of the sides. So the solution in my book: Let the sides are $a-d,a,a+d$ where $a>0,0<d<a$....
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Can the numbers $8, 15, 24$ be terms of an arithmetic progression with the common difference anything other than 1?

Let the common ratio/difference of the arithmetic progression be a number d. The exercise forces $d=/=1$ (d cannot be one). Is there any sort of proof for this exercise or am I supposed to play a ...
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1answer
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Hunt for trapezoid area & arithmetic progression sum formula similarities.

When I was looking into Arithmetic progression sum formula, I found out that it is similar to Trapezoid are formula. Arithmetic ...
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3answers
72 views

The common difference is equal to the common ratio.

Four numbers are in A.P. The first, the second and the fourth are in G.P. Find the numbers if the common difference is equal to the common ratio. Let the terms of the A.P. be $a_1,a_1+d,a_1+2d,a_1+3d$ ...
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Find $a,b,c$ of a progression

The numbers $a,b,c$ are in an arithmetic progression with $a+b+c=124$ If $a,b,c$ are the third, thirteenth and fifteenth terms of an arithmetic progression, find $a,b,c$ There are no hints in the ...
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Arithmetic progressions of permutations

I can't find this question asked anywhere else, however that may be because arithmetic progressions within permutations are an existing object of study that seemed to be returned in all my searches. ...
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1answer
23 views

Largest possible common difference of an arithmetic progression given its three terms (not necessarily adjacent) [closed]

How can I find the largest possible common difference of an arithmetic progression given that its three terms (not necessarily adjacent) are $0.37$, $9$ and $\frac{71}{7}$?
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2answers
50 views

Find the first term in a geometric progression

The exercise reads as follows: The sum of the first 5 terms in a geometric progression is 62. The 5th, 8th and 11th term of this geometric sequence are also the 1st, 2nd and 10th term of an ...
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2answers
49 views

Find the numbers $a,b,c,d$ in a geometric sequence, knowing that $a+1$, $b+6$, $c+6$, $d-4$ are in an arithmetic sequence

The exercise reads as follows: Find the numbers $a,b,c,d$ in a geometric sequence, knowing that $a+1$, $b+6$, $c+6$, $d-4$ are in an arithmetic sequence. I am interested in finding out the steps I ...
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58 views

Question on how the sum of Arithmetic Progression is $\frac{n^3}{3}$-$\frac{n^2}{2}$+$\frac{n}{6}$.

Given a set S = $({1,2,...,(n-1))}$, we obtain the sum of all of its entries through $$S_n= \frac{1}{2}n(n-1)$$ that we derive from $S_n=\frac{1}{2}n(a_1 +a_n)$. The issue is, let us square our set, $...
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1answer
39 views

Find the minimum value of 6a.

If $4\sin^2x + \csc^2x , a , \sin^2 y+ 4 \csc^2 y$ are in arithmetic progression, then find the minimum value of $6a$. My Work: Well just at a glance this problem seems to be related to AM-GM ...
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5answers
60 views

The $2$nd, the $1$st and the $3$rd term of an arithmetic progression form a geometric progression

An arithmetic progression is given with a common difference $\ne0.$ The $2nd$, the $1st$ and the $3rd$ term of the ap form a geometric progression. Find the common ratio. So we have the ap: $a_1,a_1+d,...
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Arithemetic Sequence Problem

Q: Domestic bees make their honeycomb by starting with a single hexagonal cell, then forming ring after ring of hexagonal cells around the initial cell, as shown. The numbers of cells in successive ...
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Finding the first common number in multiple sequences

I wanna know if there is a formula/algorithm to get the first common number from multiple sequences. For example: S1 = 2, 4, 6, 8, 10, 12, ... S2 = 3, 6, 9, 12, 15, ... S3 = 4, 8, 12, 16, ... First ...
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1answer
81 views

Problem with a Arithmetico-Geometric Series

Good afternoon to everyone, I have the following question: What does the arithmetico-geometric series: $$S = \sum^{\infty}_{n=1} ne^{-nrt}$$ Converge to? ($r > 1$, $t > 1$) I tried to break it ...
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1answer
22 views

Is it possible to combine an arithmetic an geometric sequence, but first adding and then multiplying?

picture of solution for when you multiply first and add after link of original discussion of the picture: Combined geometric and arithmetic series partial sum In this picture, you can see a way of ...
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Progressions of the same color in a set of consecutive natural numbers

The problem states the following: We color the first $\frac{n^3 + 5n}{6}$ numbers with red and blue. Prove that we can find $n$ numbers $a_1 < a_2 < ... < a_n$, colored with the same color ...
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20 views

Summation of terms based on given data

$ \sum_{k=1}^n {\delta}_k= 3^n-1$ then $\sum_{k=1}^\infty \frac1{\delta_k}= $ I tried writing the reciprocal terms and taking lcm but it isnt feasible since there are infinite terms
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28 views

Prove that, the ratio of two consecutive terms of an arithmetic series converges at 1

How would one prove that in an arithmetic series, as $n$ approaches infinity, the ratio of the $n^{th}$ and the $(n-1)^{th}$ term approaches $1$? [That basically means the bigger the terms of an ...
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1answer
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General formula for a special matrix

There is a special matrax $A$ (You can click the link to see the matrix): A[1][j] = 1, for $1\le j \le n$ A[i][j] = 0, for $2\le i \le n$, $1\le j \le i - 1$ A[i][j] = A[i][j-1]+A[i-1][j], ...
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for every K>1 and natural there is an arithmetic progression with coprime numbers, can't prove this

I have the following question: Prove that for every $ K \in \mathbb{N} , K\gt 1$ there exists an arithmetic progression with length K of co prime numbers. the numbers themselves aren't necessarily ...
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89 views

How do I find the first term in an arithmetic sequence?

The last term of an arithmetic sequence is $63$, the common difference is $5$ and the sum is $426$. How do you find the first term? That's all the information I have. My attempt: $63 = a + (n-1)(5)$ $...
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GP and combinations divisibility

Is the number : ${100 \choose 0}\cdot2^0 + {100 \choose 1}\cdot2^1+{100 \choose 2}\cdot2^2+{100 \choose 3}\cdot2^3 \ldots + {100 \choose 100}\cdot2^{100}$ divisible by 3? I tried looking at the first ...
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“Stronger” form of Dirichlet's theorem on primes in arithmetic progressions

Let $a$ and $q$ be two relatively prime positive integers. If I know that $$\lim_{s\rightarrow 1^{+}}\sum_{p\equiv a \bmod q}\frac{1}{p^s}=+\infty,$$ then it is clear that it tells me that there are ...
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1answer
19 views

Sequence of positives that is expressed as $3n$ [duplicate]

This is a recurrence relation that my solution as a pic maybe the right one but im looking to confirm .
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4answers
215 views

Show that three numbers form an arithmetic progression

The numbers $a,b$ and $c$ form an arithmetic progression. Show that the numbers $a^2+ab+b^2,a^2+ac+c^2,b^2+bc+c^2$ also form an arithmetic progression. We have that $2b=a+c$ (we know that a sequence ...
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1answer
51 views

Modulo of an arithmetic sequence [duplicate]

I have an arithmetic sequence, $a+dn$, where $a$ and $d$ are constants and $n$ is the term number. How would I efficiently calculate solutions to the following equation, $$ (a+dn)\%b=0 $$ where $b$ is ...
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1answer
67 views

One root common to $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$

If three distinct numbers $a,b,c$ are in GP, and the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root, then which of the following statements is correct? $1.$ $d,e,f$ are in GP. $2.$ $...
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Explanation of a variable in this ~1page paper on arithmetical progressions

I'm a first year so this is purely out of curiosity. This paper https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078964/?page=1 seems to propose a better density for arithmetical-progression free sets. I ...
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Probability that the cards are in AP

Box $1$ contains three cards bearing numbers $1, 2, 3$ ; box $2$ contains five cards bearing numbers $1, 2, 3, 4, 5$ ; and box $3$ contains seven cards bearing numbers $1, 2, 3, 4, 5, 6, 7$. A card is ...
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28 views

find if any number in A.P can be divided by a given number(k).

If there is any method other than finding each number of A.P iteratively and check if it is divisible by k or not? Example : Tn = 11*n+d; k = 7; find if (Tn % k == 0) ?
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109 views

What is a good way to prove that the sum of the first n odd natural numbers is n²?

We can try to prove the statement by direct mathematical Induction technique or by using the following method: $$S=1+3+5...+(2n-1)$$ $$S=(2n-1)+(2n-3)+(2n-5)...+1$$ $$2S=2n+2n+2n...+2n$$ $$2S=2n^2$$ ...
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2answers
61 views

A conjecture regarding Arithmetic Progressions

I have a conjecture in my mind regarding Arithmetic Progressions, but I can't seem to prove it. I am quite sure that the conjecture is true though. The conjecture is this: suppose you have an AP (...
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40 views

When does a polynomial $f(x)$ generate an arithmetic sequence for consecutive values of integer $x$?

Given polynomial $f(x) = a_0 + a_1x + \dots + a_n x^n, a_n \ne 0, n \ge 2, k \in (0,n), a_k \in \{0, 1, 2, \dots\}$, under what conditions is $f(r), f(r+1), \dots f(r+n)$ an arithmetic sequence for ...

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