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An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.
If the initial term of an arithmetic progression is a 1 \displaystyle a_1 and the common difference of successive members is d \displaystyle d , then the n \displaystyle n -th term of the sequence ( a n \displaystyle a_n ) is given by:
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:
where n \displaystyle n is the number of terms in the progression and d \displaystyle d is the common difference between terms. The formula is very similar to the standard deviation of a discrete uniform distribution.
The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be found using the Chinese remainder theorem. If each pair of progressions in a family of doubly infinite arithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is, infinite arithmetic progressions form a Helly family.[10] However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression.
If we observe in our regular lives, we come across Arithmetic progression quite often. For example, Roll numbers of students in a class, days in a week or months in a year. This pattern of series and sequences has been generalized in Maths as progressions.
A progression is a special type of sequence for which it is possible to obtain a formula for the nth term. The Arithmetic Progression is the most commonly used sequence in maths with easy to understand formulas.
Definition 2: An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
An arithmetic progression (AP), also called an arithmetic sequence, is a sequence of numbers which differ from each other by a common difference. For example, the sequence \(2, 4, 6, 8, \dots\) is an arithmetic sequence with the common difference \(2\).
We can describe an arithmetic sequence with a recursive formula, which specifies how each term relates to the one before. Since in an arithmetic sequence, each term is given by the previous term with the common difference added, we can write a recursive description as follows:
If the common difference is negative, i.e. \(da_n:\) \[ a_1 > a_2 > a_3 > \cdots .\] As an example, the arithmetic progression with initial term \(1\) and common difference \( -3, \) i.e. \( 1, -2, -5, -8, -11, \dots, \) is a decreasing sequence.
In this article, we will explore the concept of arithmetic progression, the AP formulas to find its nth term, common difference, and the sum of n terms of an AP. We will solve various examples based on the arithmetic progression formula for a better understanding of the concept.
An arithmetic progression (AP) is a sequence of numbers where the differences between every two consecutive terms are the same. In this progression, each term, except the first term, is obtained by adding a fixed number to its previous term. This fixed number is known as the common difference and is denoted by 'd'. The first term of an arithmetic progression is usually denoted by 'a' or 'a1'.
For example, 1, 5, 9, 13, 17, 21, 25, 29, 33, ... is an arithmetic progression as the differences between every two consecutive terms are the same (as 4). i.e., 5 - 1 = 9 - 5 = 13 - 9 = 17 - 13 = 21 - 17 = 25 - 21 = 29 - 25 = 33 - 29 = ... = 4. We can also notice that every term (except the first term) of this AP is obtained by adding 4 to its previous term. In this arithmetic progression:
As the name suggests, the first term of an AP is the first number of the progression. It is usually represented by a1 (or) a. For example, in the sequence 6, 13, 20, 27, 34, . . . . the first term is 6. i.e., a1 = 6 (or) a = 6.
The general term (or) nth term of an AP whose first term is 'a' and the common difference is 'd' is given by the formula an = a + (n - 1) d. For example, to find the general term (or) nth term of the progression 6, 13, 20, 27, 34,. . . ., we substitute the first term, a1 = 6, and the common difference, d = 7 in the formula for the nth term formula. Then we get, an = a + (n - 1) d = 6 + (n - 1) 7 = 6 + 7n - 7 = 7n -1. Thus, the general term (or) nth term of this AP is: an = 7n - 1. But what is the use of finding the general term of an AP? Let us see.
Therefore, the 102nd term of the given AP 6, 13, 20, 27, 34, .... is 713. Thus, the general term (or) nth term of an AP is referred to as the arithmetic sequence explicit formula and can be used to find any term of the AP without finding its previous term.
The common difference is the difference between every two consecutive terms in an arithmetic progression. Therefore, you can say that the formula to find the common difference of an arithmetic sequence is: d = an - an - 1, where an is the nth term in the progression, and an - 1 is the previous term.
A real-life application of arithmetic progression is seen when you take a taxi. Once you ride a taxi you will be charged an initial rate and then a per-mile or per-kilometre charge. This shows an arithmetic progression that for every kilometre you will be charged a certain fixed (constant) rate plus the initial rate.
When the number of terms in an AP is infinite, we call it an infinite arithmetic progression. For example, 2, 4, 6, 8, 10, ... is an infinite AP; etc. The sum of an infinite arithmetic progression doesn't exist.
The 'nth' term in an AP is a formula with 'n' in it which enables you to find any term of a progression without having to go up from one term to the next. 'n' stands for the term number so to find the 50th term we would just substitute 50 in the formula an = a+ (n - 1)d in place of 'n'.
To find d in an arithmetic progression, we take the difference between any two consecutive terms of the AP. It is always a term minus its previous term. An alternative way to find the common difference is just to see how much each term is getting added to get the next term.
This isn't precisely what was asked in the email, but it's closely related and would enable me to give a good answer. A result of Bourgain shows that if you take two dense subsets A and B of 1,2,...,n then A+B must contain an arithmetic progression of length $\exp(c(\log n)^1/3)$ or thereabouts. In particular this is true of A-A (since it contains arithmetic progressions of the same length as A-(n+1-A)). But what bounds can one get in the A-A case if one insists that the progression should be homogeneous? That is, suppose that A is a subset of 1,2,...,n of density δ. How large an m can we guarantee to find such that there exists d such that all of -dm, -(d-1)m, ... , dm are elements of A-A?
By Szemerédi's theorem applied to A, m at least tends to infinity with n and can be taken to be n logged a few times. But can we do a lot better than this? Another small observation is that if we apply Bourgain's theorem to A-A, we can obtain a quite long homogeneous arithmetic progression in A+A-A-A.
It's been a little while since I looked at either Bourgain's proof or a subsequent improvement by Green to $\exp(c\sqrt\log n)$, so I can't instantly say whether their arguments would give one a homogeneous progression in the case that B=-A. Based on my hazy memory, it feels as though it could go either way.
It's a bit late, but let me point out that there is a wonderfully short and elementary argument of Croot, Ruzsa and Schoen that gives a homogeneous arithmetic progression of length about $\log n$ in $A-A$: it can be found in a paper called Arithmetic progressions in sparse sumsets, available at ecroot/kterm.pdf.
I'm trying to represent an array of evenly spaced floats, an arithmetic progression, starting at a0 and with elements a0, a0 + a1, a0 + 2a1, a0 + 3a1, ...This is what numpy's arange() method does, but it seems to allocate memory for the whole array object and I'd like to do it using an iterator class which just stores a0, a1 and n (the total number of elements, which might be large).Does anything that does this already exist in the standard Python packages?I couldn't find it so, ploughed ahead with:
Idea 1. Given any two elements of $L$, it is easy to extend them to the maximum-length arithmetic progression containing those two elements. In particular, suppose we have $x,y \in L$ with $x
Idea 2. Say that an arithmetic progression has gap $d$ if the difference between consequence elements of the arithmetic progression is $d$. If $L$ contains an arithmetic progression of length $\ge N/4$, then its gap $d$ must satisfy $d \le 4(L[N-1]-L[0])/N$.
Idea 3. Suppose there is an arithmetic progression of length $\ge N/4$, and let $\alpha,\beta$ denote the start and end index (in $L$) of the subsequence. Since the subsequence has to be of length $\ge N/4$, the interval $[\alpha,\beta]$ must contain one or more of the following numbers: $0.2N$, $0.4N$, $0.6N$, $0.8N$ (rounding all of them to an integer).
Look at the longest arithmetic progression found at any point above. If it has length $\ge N/4$, then yes, there exists an arithmetic progression of length $\ge N/4$. If its length is $< N/4$, then no, there is no arithmetic progression of length $\ge N/4$.
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