# 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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### Logical progression of division?

A while back, I was introduced to the concept of tetration with a preface of going from simple addition (a + 1 b number of times) through multiplication (a + a b number of times), exponentiation (a * ...
1 vote
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### Arithmetic progression and proof by induction/contradiction

Show that no cube of an integer can be expressed as $7n + 5$ for some positive integer $n$ This is from Riley's "Mathematical methods for Physics and Engineering", and is question 1.28 b, ...
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### Finding or parametrizing integer solutions to $pq(p^2-q^2)=rs(r^2-s^2)$

Background: The order-3 magic square of squares problem (MSS3) is a well-known open problem that involves finding eight separate arithmetic progressions of three squares (APSs). In particular, two ...
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### Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$, given that the roots form an arithmetic progression.

Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$, given that the roots form an arithmetic progression. What I've tried : Letting $\alpha$, $\alpha+d$, $\alpha+2d$, $\alpha+3d$, $\alpha+4d$ ...
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It is known that finding 3 evenly spaced ones and 3SUM applied to (sorted) arrays with elements in the range $[0, \dots, N]$ can be solved in $\mathcal{O}(n + N \log N)$ time. Methods for solving ...