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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How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series: $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl(…
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Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is known as "The sum of the first $n$ positive integers".
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How to find a general sum formula for the series: 5+55+555+5555+.....?

I have a question about finding the sum formula of n-th terms. Here's the series: $5+55+555+5555$+...... What is the general formula to find the sum of n-th terms? My attempts: I think I need to separate 5 from this series such…
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How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result without using PMI. Is there any geometric solution to prove…
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Infinitely many primes of the form $4n+3$

I've found at least 3 other posts$^*$ regarding this theorem, but the posts don't address the issues that I have. Below is a proof that for infinitely many primes of the form $4n+3$, there's a few questions I have in the proof which I'll mark…
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Prove that there are infinitely many primes with $666$ in their decimal representation without Dirichlet's theorem.

A satanic prime is a prime number with $666$ in the decimal representation. The smallest satanic prime is $6661$. Prove that there are infinitely many satanic primes. I used Dirichlet's theorem for the progression $10000n+6661$ and it is done. I'm…
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Arithmetic and geometric sequences: where does their name come from?

Where does the name of these two famous types of sequences come from? The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the geometric mean of its two adjacent terms. Though it…
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There cannot be an infinite AP of perfect squares.

I could not find any existing questions on this site stating this problem. Therefore I am posting my solution and I ask for other ways to prove this theorem too. The Question Prove that there cannot be an infinite integer arithmetic progression of…
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$\sum \cos$ when angles are in arithmetic progression

Possible Duplicate: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\alpha + (n-1)\beta] = \frac{\cos(\alpha +…
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Elementary proof for infinitude of primes in an arithmetic progression of a special form

In this recent question the asker was looking for a proof of the existence of infinitely many prime numbers $p$ such that both $p-2$ and $p+2$ are composite. A highly upvoted answer by Ege Erdil made the point that all the primes of the form…
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Prove that $1 + 4 + 7 + · · · + 3n − 2 = \frac {n(3n − 1)}{2}$

Prove that $$1 + 4 + 7 + · · · + 3n − 2 = \frac{n(3n − 1)} 2$$ for all positive integers $n$. Proof: $$1+4+7+\ldots +3(k+1)-2= \frac{(k + 1)[3(k+1)+1]}2$$ $$\frac{(k + 1)[3(k+1)+1]}2 + 3(k+1)-2$$ Along my proof I am stuck at the above section…
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Arrange all numbers from 1 to n such that no 3 of them are in Arithmetic Progression

Is there any possible arrangement of numbers all from $1$ to $n$ such that in the resultant array of numbers, no subsequence of length $3$ is in Arithmetic Progression. For example, in $1,3,2,4,5$, there is a subsequence of length $3$ that is in AP,…
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If $a+b+c=3$, find the greatest value of $a^2b^3c^2$.

If $a+b+c=3$, and $a,b,c>0$ find the greatest value of $a^2b^3c^2$. I have no idea as to how I can solve this question. I only require a small hint to start this question. It would be great if someone could help me with this.
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If there is one perfect square in an arithmetic progression, then there are infinitely many

Consider the following positive integers: $$a,a+d,a+2d,\dots$$ Suppose there is a perfect square in the above list of numbers. Then prove that there are infinitely many perfect square in the above list. How can I do this? At first I started in this…
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Dirichlet's theorem on primes in arithmetic progression

Is there a proof in the spirit of Euclid to prove Dirichlet's theorem on primes in arithmetic progression? (By the spirit of Euclid, I mean assuming finite number of primes we try to construct another number which has a prime factor which falls in…
user17762
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