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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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Prouhet-Thue-Morse sequence and Arithmetic Progressions

This question is a fragment from a question posted by @Mathphile for which the link will be provided below. Let $(x_i)$ be an arithmetic progression of length $M$. Let $(t_i)$ be the $i$th element in ...
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Expression for the sum of another bounded series with arithmetic and geometric components

The expression I have is the following: $S = \sum_{i=1}^T (1-q)^{i-1}i$ My expression after I play around with it is: $S= \frac{1-(1-q)^TT}{q}+\frac{1-q}{q} \left(\frac{1-(1-q)^{T-1}}{q} \right) $...
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Arithmetic sequence questions

Consider the arithmetic sequence 34252, 34235, 34218,... Find that last positive term in the sequence. Since the sequence is decreasing so the formula is $a_n=-17n+34269$ for finding the nth term ...
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Find the ratio $p:q:r$, if $p,q,r$ are in H.P, and their squares are in A.P.

If three unequal numbers $p,q,r$ are in H.P and their squares are in A.P, then find the ratio $p:q:r$ . Attempt A.P(1): $\dfrac{1}{p},\dfrac{1}{q},\dfrac{1}{r}$ $$ \dfrac{1}{q}-\dfrac{1}{p}=\...
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Relation between two series

Consider the two series , A=Σ(2ⁿ/n!) from 1 to ∞. and, B=Σ(4ⁿ/n!) from 1 to ∞. What is the relationship between them?( If any) I think the exponential series might come in handy but the numerator ...
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Proof that for coprime $a$ and $b$, there is a prime of the form $an+b$

Suppose , $a$ and $b$ are coprime positive integers. Is there an easy way to show that $an+b$ is prime for some positive integer $n$ ? Dirichlet's theorem states that there are infinite many ...
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$N$ birds are distributed on a telephone wire

$N$ birds are distributed on a telephone wire that can fit a maximum of $2N$ birds. The spacings between birds form a sequence $S$. The minimum space between birds is $1$ unit. The sequence is ordered ...
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If $x, y, z$ are three distinct positive integers, where $x + y + z = 13$ and $xy, xz, yz$ form an arithmetic…

If $x, y, z$ are three distinct positive integers, where $x + y + z = 13$ and $xy, xz, yz$ form an increasing arithmetic sequence, what is the value of $(x + y)^z$ ? I've been trying to solve this by ...
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Summation of a Progression

The Progression is: $0(n-0) + 1(n-1) + 3(n-2) + 6(n-3) + 10(n-4) + .... $ which can be represented as: $\sum_{i = 0}^n \frac{i(i+1)}{2}(n-i)$ Is there a general formula for this summation?
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How to solve this variation of nim that has division?

I ran into this problem, that consist of two stacks of coins each with different amount of coins, there are two players p1 and p2. p1 plays first and each take one turn. The turn consist of removing ...
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$\sum_{r=1}^n \frac{1}{\sqrt{a+rx} + \sqrt {a+(r-1)x}}$ [closed]

Find the value of $$\sum_{r=1}^n \frac{1}{\sqrt{a+rx} + \sqrt {a+(r-1)x}}$$
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Quadratic equation including Arithmetic Progression

For $a, b, c$ are real. Let $\frac{a+b}{1-ab}, b, \frac{b+c}{1-bc}$ be in arithmetic progression . If $\alpha, \beta$ are roots of equation $2acx^2+2abcx+(a+c) =0$ then find the value of $(1+\alpha)(1+...
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If p arithmetic means are inserted

If $p$ arithmetic means $A_1$, $A_2$, ..., $A_p$, are inserted between $5$ and $41$ so that the relation is satisfied: $$ \frac{A_3}{A_{(p-1)}}=\frac{2}{5} $$ Find the value of $p/11$ I've tried ...
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Properties of this notion of density in $\Bbb{N}$?

For a given $S \subseteq \Bbb{N}$, the asymptotic density of $S$ is defined as $$d_\text{asy}(S) := \lim_{n \to \infty} {\#(k \in S : k \le n) \over n}$$ If the limit exists. Wikipedia says this is ...
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Erdos conjecture on arithmetic progression

My motivation for this enquiry is due to interest on Erdos conjecture on arithmetic progression. (https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions) Having come across ...
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Showing that the congruum is divisible by 24

Let $a,b,c \in \mathbb{N}$ be four natural numbers satisfying $b^2-a^2=c^2-b^2$. That is, $a^2, b^2, c^2$ are three successive squares in an arithmetic progression. Show that $24$ divides $b^2-a^2$. ...
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Solving $P+P=Q$ on the curve $y^2=x^3-n^2x$

If $P=(x_0,y_0)$ is a rational point on the curve $y^2=x^3-n^2x$, let $Q=2P=P+P=(x_1,y_1)$. Then $x_1$, $x_1+n$ and $x_1-n$ are all rational squares (see for example Ch 1 of the book Elliptic curves ...
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Powers progression

I know about arithmetic and geometric progression, but I thinking about progression of powers. So we have simple powers table: ...
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Finding the ratio of 5th of two different arithmetic sequences

It is given that the ratio of the sum to the nth term of two different arithmetic sequences is $7n+2:n+3$. Find the ratio of the 5th term of the sequences. I have no idea where to start this pls help!
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Expression of the sum of a power series

I was working on a mathematical stats problem and I don't get this part (that comes from a recursive arithmetico-geometric series): $$U_{k} = \alpha C^{2} + \alpha^{2} C^{2} + \alpha^{3} C^{2} + ... +...
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Let $a^2,b^2,c^2$ be natural numbers in an arithmetic progression with difference $k$.show that $24\mid k$

I need to solve the following question: Let $a^2,b^2,c^2$ be natural numbers in an arithmetic progression, with difference $k$. Show that $24\mid k$. I'm just learning the basics of number theory ...
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Evaluating a (probably) arithmetic progression

Recently, I've stumbled upon an equation (9th grade) that I know nothing about. It looks like this: ${\frac {1} {\sqrt {5}+ \sqrt {2}}}+{\frac {1} {\sqrt {8}+ \sqrt {5}}} +{\frac {1} {\sqrt {11}+ \...
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1answer
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Proof regarding arithmetic sequence

I'm given the first term $a$ and the common term $d$. I need to find an index $n$, so that the sum of first terms is $S(n)=v$. Basically, I need to solve the following quadratic system in terms of $n$...
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What kind of sequence/progression is this? What will be the answer to the question?

There are 20 urns such that the first urn contains 5 balls, the second contains 10 balls and in general the $k^{th}$ urn contains $2k + 1$ balls more that that in $(k - 1)^{th}$ urn. Then what is the ...
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5 numbers in AP, GP and HP.

The Question: Consider 5 numbers $a_1,a_2,a_3,a_4,a_5$ such that $a_1,a_2,a_3$ are in AP, $a_2,a_3,a_4$ are in GP and $a_1,a_4,a_5$ are in HP. Then find whether $\ln a_1,\ln a_3, \ln a_5$ are in AP,GP ...
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Sum of arithmetic progressions

There is this sum: $$\sum_{i=0}^{n-1}\left(\sum_{j=i+1}^{n-1}(n-j-1)\right)=\frac{1}{6}(n-2)(n-1)n$$ I don't understand how the formula is derived. What I do currently understand is this: For each i,...
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Find the number of terms in the arithmetic sequence $6 + 10 + 14 + \cdots + (4n-2)$

I am trying to understand how to find the numbers of terms in the arithmetic sequence below. $6 + 10 + 14 + ... + (4n-2)$ I think it should be $(n-1)$ because $$\frac{(4n-2)-6}{4} = \frac{4n-8}{...
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To find common difference of a logarithmic AP.

Ques: If $a,b,c$ are in GP and $\log_ba,\log_cb,\log_ac$ are in AP. Then find the common difference of AP. Here's what I did: $\Rightarrow b^2=ac$ $\Rightarrow 2\log b=\log a+\log c$ i.e. $\log a,\...
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Trading stock arithmetic progression

I started to play with a stock prediction app, and I made some table (AP) to estimate profit, but I'm not sure it is right. The app consists in predict if values of assets will rise or fall in some ...
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Arithmetic Progression (question find the sum of n terms) [closed]

Find the sum of the n terms: $$\left( 4-\frac 1n\right)+\left( 4-\frac 2n\right)+\cdots +\left( 4-\frac nn\right)$$ Note: the answer is meant to be $$\frac {7n-1}2$$
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An algebra question based on sequences and series

The following question was asked in JEE Advanced 2014 Paper I: A pack contains $n$ cards numbered sequentially from $1$ to $n$. Two consecutive cards are removed from the pack and the sum of the ...
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Find the original three terms

$x$, $y$, and $\frac3{2x}$ are non-zero terms in an arithmetic progression. If the third term is increased by $1$, the three terms now form a geometric progression. Find the original three terms. ...
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Polynomials taking a prime or 1 value on infinitely many points are irreducible

Let $ P \in \mathbb{Z}[X] $ monic of degree d such that there exists an infinite sequence $ (x_i) \subset \mathbb{Z} $ , where $ |P(x_i)| $ is prime or equal to 1. Show that P is irreducible and that ...
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An arithmetic progression of logarithms [duplicate]

$1, \log_yx, \log_zy, -15\log_xz$ are in arithmetic progression, then which of the following are correct: $z^3 = x$ $x = y^{-1}$ $z^{-3} = y$ $x = y^{-1} = z^3$ I tried converting the logs into a ...
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Arithmetical progression vs. Functions

I have been studying a lot of Maths recently but I am still in the fundamentals. Although I don't work directly with some topics, I find it very interesting. Anyways, I was studying arithmetical ...
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How do I prove this relationship between positive terms of a G.P.?

$a$, $b$, $c$, and $d$ are positive terms of a G.P. This is the relationship I'm trying to prove: $$\frac1{ab} + \frac1{cd} > 2 \left(\frac1{bd} + \frac1{ac} - \frac1{ad}\right)$$ This ...
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1answer
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How to solve for x when function can't be inverted

The sum of the first three members of a geometric progression is equal to 42. Those same members are, correspondingly, the first, the second and the sixth member of an arithmetic progression. The task ...
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How do I find the common difference if the first term is unknown?

Given the formula: $A_n = A_1+ (n-1)d$ I'm trying to look for $d$, if given the the second and $17^{\text{th}} $ terms, namely $37$ and $82$. I can't seem to figure out where to start; if $A_1$ were ...
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Arithmetic Progressions of three squares

I was reading a pdf (Arithmetic Progressions of Three Squares by Keith Conrad) and I have a question about Theorem 3.5. It says that we can use Dirichlet's theorem on primes in arithmetic progression ...
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Calculate approximate cost of service based on companies' revenue

We create bespoke web designs and we always get asked how much the designs will cost without even knowing the job. Some people don't like to be asked for their budget, some can't understand how hourly ...
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Help with how to show aritmetic progression question.

How can I show that if $(\chi_{n})$ is a aritmetic progression, then: $$\frac{1}{ \sqrt{\chi_{1}} + \sqrt{\chi_{2}} } + \frac{1}{ \sqrt{\chi_{2}} + \sqrt{\chi_{3}} } + \cdots + \frac{1}{ \sqrt{\chi_{...
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Prove by induction $(n+1) + (n + 2) +\dots + (2n - 1 ) + (2n) = \frac{n(3n + 1)}{2}$, $n \geq 1$

Prove by induction that $(n+1) + (n + 2) +\dots + (2n - 1 ) + (2n) = \dfrac{n(3n + 1)}{2}$ for $n \geq 1$. I tried to add $(2(k+1)-1) + (2(k+1))$ to both sides and got stuck. Any other ideas?
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Calculate the probability with a finite arithmetic progression

We have a finite arithmetic progression $ a_n $, where $ n \geq 3 $ and its $r\neq 0 $. We draw three different numbers. We have to calculate the probability, that these three numbers in the order ...
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Check if infinite series divisible individually by a number or not?

We are given first three numbers of a infinite series and the next elements of series would be decided by following formula: Tn= Tn-1 + Tn-2 + Tn-3 If we are ...
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Special cases of Szemeredi's Theorem?

Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial? To clarify, I'm looking for sets of integers where we can show that they contain ...
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Finding the common difference and hence, the sum of an A.P

Find the sum to $25$ terms of an A.P with the first four terms as $1, \log_yx, \log_zy,-15\log_x z$. My attempt: I started out with, $2\log_yx = 1+\log_zy$ and, $2\log_zy = \log_yx -15\log_xz$ ...
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Out of 11 tickets marked with nos. 1 to 11, 3 tickets are drawn at random. Find the probability that the numbers on them are in AP?

I kinda tried it and got the answer 5/33. But I feel like I am doing something wrong. What I did was:. Firstly I found favorable outcomes. My favourable outcomes were 25 such triplets. I think I am ...
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Find of two arithmetic progressions meet

I was working a problem to find if two AP meet or not and came across the below post Formula to find if two AP meet and in that the formula given by user is $$A_n=A_1+(n−1)d$$ $$B_m=B_1+(m−1)D$$ ...
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Compute the value of the sum $7\cdot 11+11\cdot 15+15\cdot 19+\cdots+95\cdot 99$

Solution: My attempt: =$9^2-2^2 +13^2-2^2+17^2-2^2...97^2-2^2$ =$(9^2+13^2+17^2...97^2)-2^2( 23 )$ Focus on the first part. 81+169+289+441...Here first-order differences are 88, 120, 152... and the ...
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Find number of terms in arithmetic progression

In a arithmetic progression sum of first four terms sum : $$a_1+a_2+a_3+a_4=124$$ and sum of last four terms : $$a_n+a_{n-1}+a_{n-2}+a_{n-3}=156$$ and sum of arithmetic progression is : $$S_n=350$$ $...