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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

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the common difference between consecutive terms is constant. For instance, the sequence 15, 13, 11, 9, 7, $\ldots$ is an arithmetic progression with common difference –2.

If the first term of an arithmetic progression is $a_1$, and the common difference is $d$, then the $n$th term of the sequence $(a_n)$ is $$a_n = a_1 + (n-1)d.$$

If the common difference $d$ is—

  • positive, the terms increase to positive infinity.
  • negative, the terms decrease to negative infinity.

A finite portion of an arithmetic progression is called a finite arithmetic progression or sometimes just an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

1022 questions
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Mindbending arithmetic problem

This is the question which I am referring to If the sum of $ m $ terms of an AP is equal to the sum of the next $ n $ terms of an AP as well as the sum of next $p $ terms then prove that $$(m+n)\left[\frac {1}{m}- \frac {1}{p}\right]=…
Kartik Watwani
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Summation and Arithmetic progression problem

This is the question which I am referring to If $S_n=an^2 + bn $ , verify that the series $\sum {t_{n}}$ is arithmetic where $ S_n=\sum_{n=1}^{n}{t_n} $ My try: first of all I used below equation to calculate ${t_n}$ ${t_n} = S_n - S_{n-1} = a…
Kartik Watwani
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Finding numbers and Arithmetic progression

Here is the question to which I am referring: Find three digit numbers that are divisible by 5 as well as 9 and whose consecutive digits are in AP. My way to find answer: I first arbitrarily chosen three numbers $a-d$, $a$, $a+d$ in AP where…
Kartik Watwani
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Arithmetic progression involving triplets of numbers

This is the question which I am referring to Each of two triplets of numbers (log a, log b, log c) and (log a-log2b, log 2b-log 3c, log 3c-log a) is an AP.prove that a, b, c can be lengths of sides of a triangle.also find a:b:c. My try: first of…
Kartik Watwani
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how to correctly compress expressions?

I don’t know mathematics well enough and I was faced with the task of rolling up such a thing. Could you tell me how to do this correctly? $$ \frac{3}{x^3 + 1} + \frac{5}{x^3 + 1} + ... + \frac{2x + 1}{x^3 + 1} $$ Original task
ZiEnTenIn
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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, from the section "proof by induction and…
Dimbles
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Finding the sum of the first 10 terms within an Arithmetic Progression

I have been attempting to solve the following question, however, I am unable to form any sort of relationship between the two facts. I have attempted at using all three equations, however am unable to solve anything: $T_n=a+(n-1)d$ $S_n=\frac…
Ali
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What's the value of $p$ if the roots of the biquadratic equation $x^4-10x^2+p=0$ are in AP?

What's the value of $p$ if the roots of the biquadratic equation $$x^4-10x^2+p=0$$ are in AP? The given equation is quadratic in $x^2$, so it's discriminant is $D=25-p\ge0\iff p\le25$ and the roots are $\left(x^2\right)_{1,2}=5\pm\sqrt{25-p}$. For…
yinivem462
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Confusion over wording of a question of arithmetic progression

In the highlighted part part is 15,000 given An or Sn?
Navdeep Singh
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Finding max value of $a_1 a_2 - (a_1)^2$ for an AP with 7th term 2

$a_n$ is an arithmetic progression. We are asked to find the maximum value of $a_1 a_2 - (a_1)^2$ given that its $7th$ term is $2$. This means that $a + 6d$ = $2$. From the question, $a + 6d$ = $2$. I tried taking $a_1 a_2 - (a_1)^2$ as $a(a+d) -…
FireTheLost
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How to define sets using set-builder notation where the elements of the set are in AP?

I am trying to define sets such as: $$A=\{...,-8,-3,2,7,12,17,...\}$$ $$B=\{...,-\frac{3}{2},-\frac{3}{4},0,\frac{3}{4},\frac{3}{2},\frac{9}{4},3,\frac{15}{4},\frac{9}{2},...\}$$ using set-builder notation and I'm not sure how to proceed. The…
noob anomaly
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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 case, I am asking if this intersection contain…
Safwane
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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, but don't know how to frame the function for…
Calogero
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find the arithmetic progression

find the arithmetic progression whith given $$a_2a_5=27, S_{10}=121$$ My attemp is: $$(a_1+d)(a_1+4d)=27$$ $$5(2a_1+9d)=121\rightarrow a_1=\frac{121-45d}{10}$$ Now we have the quadratic equation $$175d^2-4840d+11941=0$$ but i didnt know that is…
hi hi
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Why difference is different between terms of an arithmetic progression?

Find the common difference of the arithmetic progression $\sqrt{3}, \sqrt{27}, \sqrt{48}, \dots$ When I try to find the difference between the first and second term I end up with $2\sqrt{3}$, but when I try between the second and third terms, I end…
Astro
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