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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A polynomial f(x) of degree 10, has all its roots in A.P. with 1 being the smallest root and common diffenrence 2. then [closed]

A polynomial f(x) of degree 10, has all its roots in A.P. with 1 being the smallest root and common difference 2. then
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A combined arithmetic and geometric sequence question

Here is a question I am currently struggling with - The first, the tenth and the twentieth terms of an increasing arithmetic sequence are also consecutive terms in an increasing geometric sequence. ...
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Looking for an expansion on the AP sum formula

If I have an x where x starts at x=5, and each step adds 10, so that x1=5, x2=15, x3=25, etc...so that if there were 3 steps the answer would be 5+15+25=45. This is most properly answered by https://...
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2 votes
1 answer
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Find the values of a and b from arithmetic and geometric series

The $1^{st}$ , $2^{nd}$ and $3^{rd}$ terms of an arithmetic series are $a, b, a^2$, where $a$ is a negative number. The $1^{st}$, $2^{nd}$ and $3^{rd}$ terms of a geometric series are $a, a^2,b$. Find ...
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Common factors in two arithmetic progressions

Consider two arithmetic progressions $a+bk$ and $c+dk, $ with $a,b,c,d \in \mathbb{Z}$. Assume that for every $k \in \mathbb{Z}$ we have $$\mathrm{gcd}(a+bk,c+dk) \neq 1,$$ but a priori the gcd is ...
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Find the values $p$ and $q$ for when the geometric series converges [closed]

The numbers $p, 10, q$ are the consecutive terms of an arithmetic series. The numbers $p, 6, q$ are from a geometric series. Show that $p^2-20p+36=0$ and hence find the values of $p$ and $q$ for which ...
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1 vote
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Maximum size of subset without $k$ different three-term arithmetic progressions

Let $p \geq 3$ be a prime, and $k\geq 1$ is an integer. Suppose that $A \subseteq F^n_p$ is a subset of $F^n_p$ that does not contain $k$ different (non-trivial) three-term arithmetic progression with ...
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A specific sequence such that there is no three-term arithmetic progression in the sequence: does the corresponding series of reciprocals diverge?

Define the sequence of natural numbers $(a_n)_n$ recursively as follows: For each $k\geq 0,\ $ define $a_{k+1}$ to be the least natural number such that it doesn't make a three term arithmetic ...
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-1 votes
3 answers
123 views

Find all primes $a$, $b$, $c$ such that $a^2+a$, $b^2+b$, $c^2+c$ form an arithmetic progression

I am trying to find all primes $a$, $b$, $c$ such that $a^2+a$, $b^2+b$, $c^2+c$ form an arithmetic progression. I am curious, do any such primes even exist? If they do, can a formula to find all ...
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Sum of reciprocal powers of arithmetic series

Is there some simplifying expression for the sum $$\sum_{n=1}^\infty \frac{1}{|an+b|^x}$$ where $a,b$ are arbitrary real numbers, and $x$ is a real number larger than 1 (which should ensure ...
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1 vote
1 answer
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Prove that we can express $n^k$ as a sum of $n$ consecutive odd natural numbers

Prove that $\forall n \in \mathbb{N},$ we can express $n^k$ where $k \geq2$ is an integer, as a sum of $n$ consecutive odd numbers. My Solution - Let $2m+1$ be the first odd number , $m \geq 0 , m \...
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How to find the largest value of $n$ such that $S_n = \sum_{k = 1}^n (2k + 1) < 68$

Please help me solve the following: Given the following Sigma Notation: Given the following: $$\sum_{k=1}^n (2k+1)$$ what is the greatest number of terms for which $S_n<68$? I tried the following: $...
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Triangle of Fibonacci

I was reading in a book a presentation about the arithmetic triangle of Fibonacci (to me it also looks like the pascal triangle). The figure presented is as follows: The text says: Having arranged ...
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Is it possible to find a certain co-prime number in this arithmetic progression?

Let, $n \in \mathbb{N}$ and $(a_2,\dots,a_{n-1}) \in \mathbb{Z}$. $$\alpha_k = (1+2n + a_2n(n-1) + a_3n(n-1)(n-2) + \dots + a_{n-1}(n(n-1)\dots4*3) ~ ) + kn!, ~ ~ k \in \mathbb{Z}.$$ Is it possible ...
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3 answers
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No infinite arithmetic progression exists with prime numbers

I am trying to prove there is no infinite arithmetic progression involving only prime numbers. (In other words, I want to prove that if $a, b \in \mathbb{N}$, then there exists some $n$ such that $a + ...
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Prime numbers and Dirichlet's theorem

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ be two sequences of positive integers such that $g.c.d(a_n, b_n)=1, \ \forall \ n\geq 1$. For each $n\geq 1$, by Dirichlet's theorem, we know that there is ...
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MCQ about the product of $4$ consecutive odd numbers

The product of four consecutive odd numbers must be … (A) A multiple of 3, but not necessarily of 9. (B) A multiple of 5 . (C) A multiple of 7. (D) A multiple of 9. (E) A multiple of 3×5×7×...
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2 votes
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Geometric sequence $a,b,c,d$ and arithmetic sequence $a, \frac{b}{2},\frac{c}{4}, d-70$

The first four terms, given in order, of a geometric sequence $a,b,c,d$ and arithmetic sequence $a, \frac{b}{2},\frac{c}{4}, d-70$, find the common ratio $r$ and the values of each $a,b,c,d$. What I ...
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2 votes
2 answers
93 views

How can I derive the general formula for the standard deviation of an arithmetic series?

According to this wikipedia article, the standard deviation of an arithmetic series has the following formula, in terms of the common difference $d$ and number of terms $n$. The article does not cite ...
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1 vote
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62 views

Proportionality constant

I’ve typed the question, in whose context my doubt is, and it’s answer at the end. Please note that I do not require the solution as I’ve already understood how to find the answer via the given as ...
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Primes in coupled arithmetic progressions

Dirichlet's theorem on arithmetic progressions tells us that there are infinitely many primes in an arithmetic progression $a+nd$ for coprime positive integers $a,d\in\mathbb{N}$ fixed and $n\in\...
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Simultaneous Equations with coefficients in an arithmetic sequence

Solve: $2x+5y=8, -10x-7y=-4$ or $x+2y=3, 4x+5y=6$ you will find that they have the same solutions. In fact any two simultaneous equations of the form: $ax+(a+d)y=a+2d, bx+(b+e)y=b+2e$ will have the ...
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2 votes
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56 views

Condition for two quadratic equation to have one common root (Simplification)

If a,b,c are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx2+2ex+f=0$ have a common root if $\frac da, \frac eb, \frac fc$ are in: Arithmetic Progression Geometric Progression ...
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how to find limit of Arithmetic progression equation

i try to find limit of equation, where d!=0, $$ \lim_{x\to\infty} \frac{S_n}{a^2_n} $$ i use some formulas to find limit, but it's give me nothing, irreducible fraction: $$ \lim_{x\to\infty} \frac{S_n}...
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How do I make effective use of the formula for calculating the product of the members of a finite arithmetic progression?

Here's a question I was tasked to answer with limited knowledge/use of combinatorics: In a class of 25 students, every student gets to pick a number 1 to 100. What is the probability that at least two ...
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Is it possible to calculate where the first point in “a_n = a_(n-1) + 2 with starting point C” equals a square (series with the same growth)?

Background: Solving this could lead to something revolutionary which is why I don't think it is possible ... but it seems feasible. Despite the seemingly simplistic question, it is actually quite ...
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Optimal reward reinvesting interval with non-zero transaction fee

Description of the system I have a system where every interval of time (lets assume every day) a reward (n) is paid out.The amount n paid out every day/interval is constant. (i.e n=20$) I have the ...
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Does there exist a subset of N satisfying a certain density property

If $A\subset \mathbb{N}$ we say that the set $A$ has a natural density if the limit $$\lim_{n \to \infty}\frac{|A\cap [1,n]|}{n}$$ exists and we denote it by $d(A)$. Also for every $k\in \mathbb{N}$ ...
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2 answers
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Existence of closed form formulas for integer sequences

Suppose that z x y ———————— 0 1 0 1 2 0 2 2 1 3 3 0 4 3 1 5 3 2 6 4 0 7 4 1 8 4 2 9 4 3 . . . . . . . . . n xn yn the z column ...
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2 votes
1 answer
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Is there an easy method to determine the amount of x solution to be added to y solution to form 1%, 5%, 10%, 20%, and 40% solutions?

I'm doing some calculations for a chemistry experiment and have decided that I need solutions of 1%, 5%, 10%, 20% and 40% consisting of x mL of ethanol and y mL of solution y. Are there any possible ...
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-3 votes
2 answers
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Find the arithmetic progression where $a_7 \cdot a_8 = 1326$ and $a_1 \cdot a_{14} = 276$ [closed]

Find the arithmetic progression where $a_7 \cdot a_8 = 1326$ and $a_1 \cdot a_{14} = 276$ Right now I've only dealt with these problems when I have a sum of two members of the AP so I can use the ...
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  • 503
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1 answer
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In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d.

Q: In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d. My solution: $S_5$ =$\frac{1}{4}$*{$S_{10}$- $S_5$}. Q says that it is ...
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1 answer
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Q regarding finding total sum at end of year with an increment till n-1th year.

Q: The income of a person is 3,00,000, in the first year and he receives an increase of 10,000 to his income per year for the next 19 years. Find the total mount, he received in 20 years. Currency = ...
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4 votes
1 answer
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Number of k-APs in Z/NZ?

Let $\mathbb{Z}/N\mathbb{Z}$ be the module class of an integer $N$. Let $k$ be an integer and we may assume $N$ is much larger than $k$ to avoid some trivial case. If necessary, we can also assume $N$ ...
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Generalised formula for the given series

I have the below series : (1 * 0) + (2 * 1) + (3 * 2) + (4 * 3) + ... + (n * (n-1)) Is it possible to have a generalised formula for this. Also such series like ...
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  • 101
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0 answers
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Generalized Euler's lucky numbers of the form $m(n^2 - n) + p$

There exist analogues of Euler's lucky numbers. It is numbers of the form $m(n^2 - n) + p$ which produces primes for $ 0 < n < p$. For example, if we take $2(n^2 - n) + 19$, then it's first 18 ...
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1 answer
119 views

Does $\mathbb{N}$ have fewer primes than any arithmetic progression?

Let $f(x):=ax+b$ for naturals $x,a,b$, such that $f(x)$ will take infinitely many primes as $x\to\infty$. Is it the case that for any choice of $a,b$, there exists some $N$ such that for all $n>N$, ...
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4 votes
0 answers
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Can we find a palindrome for more than $5$ factors?

For positive integers $a,b$ and $k$ , define $$p(a,b,k):=\prod_{j=1}^k (a(j-1)+b)$$ That is the product of the first $k$ numbers in the arithmetic progression $an+b$ starting with $b$ Can $p(a,b,k)$ ...
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-1 votes
1 answer
32 views

How to find the RHS of $\sum_{1 \leq a \leq \lfloor{\frac{n}{2}}\rfloor} (n-2a) $ [closed]

How does $$\sum_{1 \leq a \leq \lfloor{\frac{n}{2}}\rfloor} (n-2a) = \lfloor{\frac{n}{2}}\rfloor (n - \lfloor{\frac{n}{2}}\rfloor - 1)$$ I cannot see how you get to the RHS. This looks close to number ...
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0 votes
0 answers
29 views

Primes in the intersection of arithmetic progressions

Let $a,b,r,s$ be given constants. We know that that the arithmetic progressions $\{ax + r : x \in \Bbb Z\}$ and $\{by + s : y \in \Bbb Z\}$ intersect if and only if $\gcd(a, b) \mid (s − r)$. In this ...
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-1 votes
1 answer
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Prove $\frac{1}{u_{0} u_{1}}+\frac{1}{u_{1} u_{2}}+\ldots+\frac{1}{u_{i} u_{i+1}}+\ldots \ldots+\frac{1}{u_{n} u_{n+1}}=\frac{n+1}{u_{0} u_{n+1}}$ [closed]

Let $\mathrm{u}_{\mathrm{n}}$ be an arithmetic progression (sequence) with common difference $\mathrm{d}\left(\mathrm{u}_{\mathrm{i}+1}=\mathrm{u}_{\mathrm{i}}+\mathrm{d}, \mathrm{i} \geq 0\right)$. ...
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0 answers
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How to find the ith element in two merged arithmetic progressions?

Suppose we have two sequences: $a_i=ai, b_i=bi$ where $a>0, b>0, i\in\mathbb{N}, a\in\mathbb{R}, b\in\mathbb{R}$ We then define $u_i$ the result of merging the two sequences such that all ...
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  • 167
2 votes
3 answers
72 views

Proving $a_n = a_i + (n - i)d$ by induction

The formula for the $n$th term of an arithmetic sequence is $$a_n = a_1 + (n - 1)d \tag{1}\label{1}.$$ However, I want to prove that this can be generalized to $$a_n =a_i + (n - i)d. \label{2}\tag{2}$$...
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Convex sided polygon with exterior angles in AP [duplicate]

This question has been asked before, but I have doubts regarding the answer given and being accepted over there : (Link :- Convex n-sided polygons whose exterior angles expressed in degrees are in ...
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  • 633
0 votes
2 answers
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sequences and series problem - returns on investment

I am having a problem with these types of problem. Kenny is offered 2 investment plans , each requiring an initial investment of £10,000. Plan A offers a fixed return of £800 per year. - arithmetic ...
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  • 3
5 votes
2 answers
4k views

Why does the primorial $23\#$ come up so often in long prime arithmetic progressions?

This section of the Wikipedia article on the Green-Tao theorem gives examples of the longest known arithmetic progressions of prime numbers. For every known arithmetic progression of $24$ or more ...
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3 votes
1 answer
410 views

Convex n-sided polygons whose exterior angles expressed in degrees are in arithmetic progression

If the exterior angles of a convex n-sided polygon, are all integers, expressed in degrees, are in arithmetic progression, how many values are possible for $n$? The sum of all exterior angles has to ...
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0 votes
1 answer
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An arithmetic progression problem

The question was: An AP starts with a positive fraction and every alternate term is an integer. If the sum of the first $11$ terms is $33$, find the fourth term. I considered the first term to be $a$...
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1 answer
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How do I create a function that gives $2+k, 3+K$ successively for all $k$ in $\mathbb{N}$?

In doing unrelated research, I conjectured that $n^3+(n+3m)^2$ is divisible by $3$ for all n in $[2+k,3+k]$ for all $k$ in $\mathbb{N}$ for all m in $\mathbb{N}$. I'd like to prove this algebraically, ...
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3 votes
1 answer
147 views

"Magic" numbers are those divided by all partial digit sums: prove that there is no infinite set of "magic" powers among the natural powers of $\ell$

For a natural number $n$, let $P_n$ the set of sums of each subset of digits in decimal notation of $n$. A number is magic if for each $s \in P_n$, we have $s \ | \ n$. Let's consider a number $\ell$, ...
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