# 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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### 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 ...
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 ...
1 vote
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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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### 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 ...
27 views

### 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 ...
1 vote
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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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1 vote
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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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### 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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### 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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### 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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### 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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### 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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### 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 $a$...
### 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, ...
### "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 $\ell$, ...