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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Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$
Let $X:=\{$ positive integers that contain the digit $2\}$
For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$
I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap ...
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Interesting NT Question With AP and GCD.
Find the number of prime triplets $(p, q,r)$ such that $p(p + 1), q(q + 1),r(r + 1)$ form a strictly increasing arithmetic progression, where GCD $(r − p, 2p + 1)=1$.
What I tried:
$r(r+1)-p(p+1)=2d$ ...
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Quadratic where roots and coefficients together form Arithmetic Progression
Background
I was reading this post: A.P. terms in a Quadratic equation.
And wondered the following: Given a quadratic $ax^2+bx+c=0$ which has roots $x=m,x=n$, is it possible for $a,m,b,n,c$ to be ...
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How to form an AP which contains common terms of two other APs?
I got this question when I was going through a question from the infamous entrance exam JEE. It was a question from the JEE Advanced 2018, Paper 1.
The question is as follows:
Let X be the set ...
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Given $a,b,c,d$ in arithmetic progression, can we express the solution to $\frac1{x-a}+\frac1{x-b}+\frac1{x-c}+\frac1{x-d}=0$ in terms of $a,d$ only?
Context
This is a question I wrote for myself recently, heavily based on an old (1930s) examination paper for university admissions.
Given $a,b,c,d$ which are real numbers and consecutive terms in an ...
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Exercise $3.2.6$ in Tao-Vu's Book - why is it trivial for $\epsilon \geqslant 8^{-d}$?
Let $P$ be a proper progression of rank $d$ in an additive group $(Z,+)$, and let $A\subset P$ such that $|A| \leqslant ε|P|$ for some $0 <ε< 1.$ Show that $P\setminus A$ contains a proper ...
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Arithmetic Progressions involving 3 distinct terms
I've been trying to find the number of Arithmetic Progressions involving 3 distinct terms of the set:
$ \sum_{i=0}^{2021} a_i \cdot 3^i, \quad \text{where } a_i \in \{0, 2\}$
But I've had no sucess, ...
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Application of $a_n$ = $S_n$ - $S_{n-1}$ for an arithmetic progression.
Consider the following question:
Let $V_r$ denote the sum of the first r terms of an Arithmetic Progression whose first term is r and common difference is (2r-1). Let $T_r$ = $V_{r+1}$ - $V_r$ -2. ...
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$\frac{a}{x},\frac{b}{y},\frac{c}{z}$ are in HP
If non-zero numbers $a,b,c,x,y$ and $z$ are such that $a,b,c$ are in AP, $x,y,z$ are in GP and $\frac{a}{x},\frac{b}{y},\frac{c}{z}$ are in HP then prove $|a|=|c|$.
I have been trying to solve this ...
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Prove $p,q,r$ are AP if $p\left(\frac{1}{q}+ \frac{1}{r}\right), q\left(\frac{1}{r}+ \frac{1}{p}\right), r\left(\frac{1}{p}+ \frac{1}{q}\right)$ is AP
How should one approach this question. The following is my attempt:
I started by forming the following equation, which is the inherent property of Arithmetic Progression:
$$2 \cdot q\left(\frac{1}{r}+ ...
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Relationship between the squares of first n natural numbers and first n natural odd numbers.
Here's a question from high school mathematics.
If $ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + \dots + 100^2 = x $, then
($1^2 + 3^2 + 5^2 + \dots + 99^2$) is equal to ?
Options were:
(a) $\frac{x}{2}-2525$
(b) ...
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I have got stucked with this concept of A.P.
Q) How to prove that the sequence:$ 2,4,6,8,...,1000$ is an A.P.$($$Arithmetic$ $Progression$)?
First of all, the $1^{st}$ term of this sequence is $2$ and the common difference of this sequence is ...
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Formula for calculating the sum of the equation: $y = \lfloor 400(x-6)^{1.1} \rfloor$
I have an equation of $y = \lfloor 400(x-6)^{1.1} \rfloor$ where x is equal to or greater than 6 and increases by an increment of 1.
I want to calculate what the sum of the equations added up together ...
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What is the formula for finding the summation of the sequence : $1,2,5,12,26,51,...$ upto $n$ terms? [duplicate]
Q)What is the formula for finding the summation of the sequence $1,2,5,12,26,51,...$ upto $n$ terms ?
I know how to find the summation of sequences like $1,2,3,...,$ upto $n$ terms ; $1,2,4,8,...,$ ...
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Can any of you do something relevant with this mathematical property I found? [closed]
I'm an amateur mathematician and I found a property that I've never seen anyone mention before, I think I managed to demonstrate it below. I confess that I don't know if it's new or not, as I haven't ...
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