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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.

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Prove that $n$ is also a power of $2$.

An arithmetic progression consists of integers. The sum of the first $n$ terms of this progression is a power of two. Prove that $n$ is also a power of two. Source :http://www.math.ucla.edu/~radko/circles/lib/data/Handout-967-1026.pdf
Amar30657
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Centered hexagonal numbers

This is one of Brilliant's daily challenges. What I see is four arithmetic progressions. I did my calculations according to the formulas: $$S_{n1} = \frac n2(2a + (n − 1) × d_1),$$ $d_1=1$. $$ S_{n2} = \frac n2(2a + (n − 1) × d_2),$$ $d_2=2$. $X =…
Sergey Zolotarev
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A disc is cut into 12 sectors with areas in arithmetic progression. The largest angle is twice the smallest. Find the smallest angle.

I was given a question which states - A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence. The angle of the largest sector is twice the angle of the smallest sector. Find the size of the angle of the smallest…
JonDoe
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Question about arithmetic progression. stuck on one of the answers

So i was doing this question on my workbook. I did the first question and i was correct. but on the second question, same logic, same solving way, but i was wrong. here is question 1, finding a closed form for $$a_1=3\qquad…
doodler doodle
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How to find the sum of the the first n elements in the series 1,2,4,5,7,8.... as a function of n?

I see I can split it up into two arithmetic series (1,4,7... and 2,5,8...) and add the two sums, but what is the compact notation for the sum of these two series as a function of n?
Ido Sarig
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$S_n = x_1^3+x_2^3+ \cdots +x_n^3$ squares perfect

It is considered arithmetic progression $x_1,x_2, \cdots, x_n,\cdots, x_1 \neq0$ . Show that if sums $$S_n = x_1^3+x_2^3+ \cdots +x_n^3$$ is squares perfect for any natural $n \in N$, then there are $k\in N^*$ so $x_n=nk^2$, for any $n \in N$. All…
medicu
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What is the first term of the arithmetic sequence $3,7,11, \ldots$ that exceeds $200$?

The first three terms of a sequence are $3,7,11$. What is the first term to exceed 200? Here's what I've done so far: Common difference: $T_2-T_1 = 7-3=4.$ $200 + 4=204$ Therefore $204$ is the first term to exceed $200$. But when I…
AugieJavax98
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Why does $n*T_n = m*T_m$ imply $T_\text{n+m} = 0$?

I was fiddling around with Arithmetic Progressions and I noticed this pattern. \begin{align} n*T_n = m*T_m \implies T_\text{n+m} = 0 \end{align} where $n, m \in \{0, \mathbb{Z}^{+}\}$ and $n \neq m$. I could prove that it's true like…
HerrAlvé
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The correct solution for a 10th grade A.P. problem

Show that $a_1$, $a_2$,...,$a_n$,... form an AP where $a_n$ is defined as below: $a_n = 3+4n$ This is a problem from a grade 10th textbook... I solved the question in a different way from how our teacher did... My question is that was my method…
GameTime With Aryan
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How to prove that these conditions generate an arithmetic sequence?

The sequence $a_n$ satisfies $$a_1=1,\\a_n+a_{n+12}=2a_{n+6},\\a_n+a_{n+14}=2a_{n+7}.$$ I know that these conditions imply that $a_n$ consists of several arithmetic subsequences with index interval as $6$ and $7.$ Because $6$ and $7$ are coprime, I…
Hans J
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The sides of a triangle are in A.P.

One of the angles of a triangle is $120^\circ$. The sides of the triangle are in A.P. Find the ratio of the sides. \begin{align} 3:5:7 \end{align} So the solution in my book: Let the sides are $a-d,a,a+d$ where $a>0,0
Math Student
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How can I find the first n terms in an arithmetic sequence if the last term isn't given

I have two questions. Suppose I have an arithmetic sequence $11,20,29,38,...$ and $11+20+29+38+... = 8998$. How can I find the first $n$ terms that would get me to 8988? I know that the sum is $\frac{n(a_1+a_n)}{2}$ and I have $8988 =…
user130306
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Product of arithmetic progressions

Let $(a_1,a_2\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,\dots,a_nb_n)$ is an arithmetic progression? The sequence is (trivially) an…
pi66
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An arithmetic sequence with no squares

While playing with another problem, I found out that $a_n=4n+3;\;n\in\mathbb{N}$ contains no squares. I tried to prove it in this way $4n+3$ is odd so we must find an integer $m$ such that $4n+3=(2m+1)^2$ that is $$4n+3=4m^2+4m+1$$ Solving for $n$ I…
Raffaele
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Can't understand this question related to arithmetic progression.

I can't understand one thing in one question of Arithmetic Progression. I have to find $21^{st}$ term from the set $\{12,2,-4,-10\}$ and this is where problem start. Arithmetic Progression Equation: $T_n=a+(n-1)d$ $Tn$ = Term $n$ = Term Number $a$ =…
Murtaza
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