What is Sequence: Definition and 1000 Discussions

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences), or the set of the first n natural numbers (for a sequence of finite length n). Sequences are one type of indexed families as an indexed family is defined as a function which domain is called the index set, and the elements of the index set are the indices for the elements of the function image.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).
The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of




a

n




{\displaystyle a_{n}}
,




b

n




{\displaystyle b_{n}}
and




c

n




{\displaystyle c_{n}}
, where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence



F


{\displaystyle F}
is generally denoted as




F

n




{\displaystyle F_{n}}
.

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

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  1. L

    Proving convergence of sequence from convergent subsequences

    In the photos are two proof questions requiring proving convergence of sequence from convergent subsequences. Are my proofs for these two questions correct? Note in the first question I have already proved that f_n_k is both monotone and bounded Thanks a lot in advance!
  2. Dries vanlandschoot

    Does the Modified Collatz Conjecture Always End in 1, 5, or 17?

    Did some calculations and with 3×-1 i Always get 5.is this correct?
  3. kardo

    I What is the missing number in this number sequence pattern?

    these are a pattern of number sequence asked in assessment for new employee , what could be the missed number 9 1 6 4 4 5 7 2 5 8 8 5 1 3 5 X is says the equation could be limited to (addition, subtraction, division and multiplying)
  4. E

    B Pattern inductive reasoning problem

    In attempting to finish the sequence, I realized that the triangles in the inner hexagon is decolored in the direction of the arrow I drew for the given elements of the sequence. Any tips on how to think systematically about similar problems since I am required in the exam to complete like...
  5. Jiketz

    A Determining rationality of real numbers represented by prime digit sequence

    I would like to know if my answer is correct and if no ,could you correct.But it should be right I hope:
  6. chwala

    Determine Convergence/Divergence of Sequence: f(x)=ln(x)^2/x

    ##a_n= \left[\dfrac {\ln (n)^2}{n}\right]## We may consider a function of a real variable. This is my approach; ##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]## Applying L'Hopital's rule we shall have; ##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[...
  7. J

    Bounded non-decreasing sequence is convergent

    So far this is what I have. Proof: Let p1, p2, p3 be a non-decreasing sequence. Assume that not all points of the sequence p1,p2,p3,... are equal. If the sequence p1,p2,p3,... converges to x then for every open interval S containing x there is a positive integer N s.t. if n is a positive integer...
  8. H

    Monotonic sequence definition of Continuity of a function

    Question: There is a function ##f##, it is given that for every monotonic sequence ##(x_n) \to x_0##, where ##x_n, x_0 \in dom(f)##, implies ##f(x_n) \to f(x_0)##. Prove that ##f## is continuous at ##x_0## Proof: Assume that ##f## is discontinuous at ##x_0##. That means for any sequence...
  9. elcaro

    I Exploring the Expanded Collatz Sequence: Is it Unique for Each Value of k?

    Has this expanded Collatz sequence been explored previously?
  10. S

    How do the Setter dog breeds relate to the Predatory Sequence?

    Maybe this is not a well made question, but how or in what ways are the Setter dog breeds related to the Predatory Sequence? What parts of the sequence are expressed and what parts are inhibited? Interested readers who might not be much aware of the Predatory Sequence can do some simple...
  11. jedishrfu

    Great Sequence of Video Documentaries on Nuclear Disasters

    Youtuber Kyle Hill has created a great sequence of videos on nuclear disasters around the world: They include: - THERAC-25 Medical Device disaster where cancer outpatients received doses many times greater than what the doctors prescribed - The Demon Core: fissile material that scientists...
  12. H

    I If a sequence converges, then all subsequences of it have same limit

    Let's say we're given a sequence ##(s_n)## such that ##\lim s_n = s##. We have to prove that all subsequences of it converges to the same limit ##s##. Here is the standard proof: Given ##\epsilon \gt 0## there exists an ##N## such that $$ k \gt N \implies |s_k - s| \lt \varepsilon$$ Consider...
  13. H

    Proving that there is a sequence in S, such that ##\lim s_n = \sup S##

    Let ##S=\{s_n:n∈N\}##. ##\sup S## is the least upper bound of S. For any ϵ>0, we have an m such that ##\sup S−\epsilon \lt s_m## ##\sup S−s_m \lt \varepsilon## ##|\sup S−s_m| \lt \varepsilon## I mean to say that, no matter how small ϵ is, there is always an element of S whose distance from supS...
  14. H

    If ##|s_{n+1} - s_n| \lt 1/2^n##, then ##(s_n)## is a Cauchy sequence

    My attempt: It can be proved that ##\lim \frac{1}{2^n} = 0##. Consider, ##\frac{\varepsilon}{k} \gt 0##, there exists ##N##, such that $$ n \gt N \implies \frac{1}{2^n} \lt \varepsilon $$ Take any ##m,n \gt N##, and such that ##m - k = n##. ##|s_m - s_{m-1} | \lt \frac{1}{2^{m-1}} \lt...
  15. Eclair_de_XII

    B Let f_n denote an element in a sequence of functions that converges

    Let ##\epsilon>0##. Choose ##N\in\mathbb{N}## s.t. for each integer ##n## s.t. ##n\geq N##, $$|\sup\{|f-f_n|(x):x\in D\}|<\frac{\epsilon}{3}$$ where ##D## denotes the intersection of the domains of ##f## and ##f_n##. Choose a partition ##P:=\{x_0,\ldots,x_m\}## with ##x_i<x_{i+1}## where...
  16. Bob Walance

    I Finding the period in a modular exponentiation sequence

    This question is related to Shor's algorithm and its use of modular exponentiation. In the table below, the period of the sequence in the third column is obviously equal to 4. That is, its value repeats every fourth row. What I am trying to find out is why it is that when the first value in...
  17. M

    One-line Python Code that returns a sequence of integers

    Hey! 😊 I want to write a one-line Python generator or iterator expression that returns the sequence of integers generated by repeatedly adding the ascii values of each letter in the word “Close” to itself. The first 10 integers in this sequence are: 67, 175, 286, 401, 502, 569, 677, 788, 903...
  18. C

    I Proving a convergent sequence is bounded

    Dear Everybody, I have a quick question about the \M\ in this proof: Suppose \b_n\ is in \\mathbb{R}\ such that \lim b_n=3\. Then, there is an \ N\in \mathbb{N}\ such that for all \n\geq\, we have \|b_n-3|<1\. Let M1=4 and note that for n\geq N, we have |b_n|=|b_n-3+3|\leq |b_n-3|+|3|<1+3=M1...
  19. chwala

    Find the two values of the common ratio- Geometric sequence

    *kindly note that i do not have the solutions ...I was looking at this, not quite sure on what they mean by exact fractions, anyway my approach is as follows; ##\dfrac {a}{243}=\dfrac{a(1-r^3)}{240}## ##\dfrac {1}{243}=\dfrac{1-r^3}{240}## ##\dfrac {240}{243}=1-r^3##...
  20. S

    Probability of getting arithmetic sequence from 3 octahedron dice

    I try to list all the possible sequences: 1 2 3 1 3 5 1 4 7 2 3 4 2 4 6 2 5 8 3 4 5 3 5 7 4 5 6 4 6 8 5 6 7 6 7 8 I get 12 possible outcomes, so the probability is ##\frac{12 \times 3!}{8^3}=\frac{9}{64}## But the answer key is ##\frac{5}{32}## . Where is my mistake? Thanks
  21. T

    I Finding a general formula for a recursive sequence

    The general formula for the nth term of the Fibonacci sequence where an=an-1+an-2 can determined by matrix diagonalizations Is there a way to determine the formula of any recursive sequence say an=a1+a2+...an-1
  22. A

    What is the summation of this partial sequence?

    Greetings! I want to caluculate the summation of this following serie I started by removing the 4 by and then and I thought of the taylor expansion of Log(1-x)=-∑xn/n but as the 2 is not inside (-1,1) I couldn´t use it any hint? thank you! Best !
  23. WMDhamnekar

    MHB Properties of sequence of random numbers and how to generate random numbers?

    Hi, Would any member of Math Help Board explain me the highlighted area in the following paragraphs? Generating Random Distributions Now the only missing thing in previous cases is how would one generate a Uniform random, Normal random distributions. We therefore look to cover algorithms to...
  24. elleninphysics

    Intro Physics BEST Calc- and Algebra-Based Textbooks with the same Chapter Sequence?

    Hi there! I am teaching the algebra- and calculus-based physics courses at my university. The courses are taught at different times but in parallel, and students from both courses share the same lab sections. I'd like to keep them on the same content schedule without jumping around the...
  25. karush

    MHB 13.3.2 What is the 50th term of the sequence

    The 3rd and 4th terms of an arithmetic sequence are 13 and 18. respectively. What is the 50th term of the sequence! a, 248 b. 250 c. 253 d, 258 e, 763 b the common difference is 5 so $5\cdot 50=\boxed{250}$ basically these are easy but I still seem to miss the goal posts
  26. M

    B What happens to the Core of a main sequence star as additional mass is added?

    So there is a post going around the facebook groups about what would happen if we could pour a Sun sized bucket of water onto the sun, the claim being that the sun would gain mass and become a bigger, hotter burning blue star. I know this cannot happen but I was just curious as to what would...
  27. K

    Limit of a Sequence: Does Square or Sqrt Change It?

    Does the square of the sequence also have a limit of 1. Does the square root also equal 1? I've been trying to find some counterexamples but I think the limit doesn't change under these operations?
  28. C

    I What does "the sequence of functions has limit in R" mean?

    Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?
  29. chwala

    Find the nth term of the given sequence

    My working...i need to arrange this in an array, ##[2, 9, 28, 65, 126]## ##[ 7, 19, 37, 61 ]## ##[ 12...
  30. C

    I Factoring Matrices with Elementary Row Operations

    Dear Everybody, I have some trouble with this problem: Finding a sequence of elementary matrix for this matrix A. Let ##A=\begin{bmatrix} 4 & -1 \\ 3& -1\end{bmatrix}##. I first used the ##\frac{1}{4}R1##-> ##R1##. So the ##E_1=\begin{bmatrix} \frac{1}{4} & 0 \\ 0& 1\end{bmatrix}##. So the...
  31. A

    Straight line intersects geometric sequence

    Summary:: Two parallel lines (same slope) - one intersects the y-axis, and the other doesn't. Trying to find the intersection of either with a given geometric sequence. The lines are: y=mx y=mx+1 The values on one or the other of the lines - but not both simultaneously - are to be completely...
  32. M

    MHB Extrema and convergence of sequence

    Hey! :giggle: For $n\in \mathbb{N}$ let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ given by $f_n(x)=\frac{x+2n}{x^2+n}$. (a) Determine all (local and global) extrema of $f_n$ and the behaviour for $|x|\rightarrow \infty$. Make a sketch for $f_n$ and $f_n'$. Show that there exists $x_1<x_2<x_3<x_4$...
  33. Eclair_de_XII

    B Question about set containing subsequential limits of bounded sequence

    Let ##L\in E##. By definition, there is a subsequence ##\{x_{n_k}\}_{k\in\mathbb{N}}## that converges to ##L##. There is a natural number ##N## s.t. if ##n_k\geq N##, ##L\in(x_{n_k}-1,x_{n_k}+1)\subset(\inf\{x_n\}-1,\sup\{x_n\}+1)##. Hence, ##E## is a bounded set. If ##E## is a finite set, then...
  34. Eclair_de_XII

    B Convergence of a sequence of averages of a convergent sequence

    Let ##\epsilon>0##. Then there is an integer ##N>0## with the property that for any integer ##n\geq N##, ##|a_n-A|<\epsilon##, where ##A\in\mathbb{R}##. If for all positive integers ##n##, it is the case that ##|a_n-A|<\epsilon##, then the following must hold: \begin{eqnarray}...
  35. F

    Prove that the sequence does not have a convergent subsequence

    Hello i have problems with this exersice Let $$\{X_{\alpha}\}_{\alpha \in I}$$ a collection of topological spaces and $$X=\prod_{\alpha \in I}X_{\alpha}$$ the product space. Let $$p_{\alpha}:X\rightarrow X_{\alpha}$$, $$\alpha\in I$$, be the canonical projections a)Prove that a sequence...
  36. R

    Combinatorics question about the four-letter sequence "GRIT"

    Question: "A total of 9! = 362880 different nine-letter ‘‘words’’ can be produced by rearranging the letters in FULBRIGHT. Of these, how many contain the four-letter sequence GRIT?" Solution: There are six ways of getting the word "GRIT" with five letters left over giving 6 x 5! = 720...
  37. malawi_glenn

    Help with a proof regarding convergent sequence (proof by contradiction)

    Ok I am trying to brush up my real analysis skills so that I can study some topology and measure theory at some point. I found this theorem in my notes, that is proven by using proof by contradiction. However, I have a hard time understanding what the contradiction really is... Here is the...
  38. T

    Probability of Receiving 2-4-6-7-Queen-King Sequence from 40 Cards

    A quick translation of the question: We have a deck of 40 cards, containing four suits (hearts, diamonds, spades, clubs), in which each suit has an ace, a queen, a jack, a king and six number cards (2 to 7). From the deck, six cards are distributed randomly and successively to a player who picks...
  39. L

    Recursive sequence calculators

    I know the http://www.calcul.com/ but it is offline some days. Do you know any other online free ones? I.e. that calculates the far values of a recursive sequence.
  40. S

    MHB Limit of a Sequence (Updated with progress)

    Hello, I know I posted this question recently but I wanted to update with my progress. I have figured out what the limit should be but I would really appreciate help with how to use the definition of the limit of a sequence to prove it! What I have is:Suppose n is extremely large, then both...
  41. S

    MHB Discover the Limit of a Sequence with Easy-to-Follow Steps

    Hello! I have been trying to work through this but I have never really been able to use the definition correctly to find a limit sequence. Any help would be greatly appreciated!
  42. anemone

    MHB Sequence Challenge: Prove $a_{50}+b_{50}>20$

    The sequence $\{a_n\}$ and $\{b_n\}$ are such that, for every positive integer $n$, $a_n>0,\,b_n>0,\,a_{n+1}=a_n+\dfrac{1}{b_n}$ and $b_{n+1}=b_n+\dfrac{1}{a_n}$. Prove that $a_{50}+b_{50}>20$.
  43. Eclair_de_XII

    B Continuation of sequence of supremums question

    First, it must be justified that ##A_n## is decreasing and is bounded from below by the point it converges to. See the other topic (1) ##A_n## is decreasing By definition, ##A_n## is the supremum of the set containing all ##a_k## where ##k\geq n+1## and the set containing ##a_n##. Hence, it...
  44. Eclair_de_XII

    B Showing that a sequence of supremums of a sequence has these two properties

    ===(1)=== Let ##n\in \mathbb{N}##. Express ##A_n## and ##A_{n+1}## as: ##A_n=\sup\{a_n,a_{n+1},\ldots\}## ##A_{n+1}=\sup\{a_{n+1},\ldots\}## Suppose for some ##m\geq {n+1}##, ##a_m=A_{n+1}##. By definition, ##a_m\geq a_k## for ##k\geq {n+1}##. If ##a_n<a_m##, then ##a_m\geq a_k## for ##k\geq...
  45. S

    MHB Is the Sum of n^2 Terms in an Arithmetic Sequence Limited to 1?

    How many different arithmetic sequences have the sum of the first n terms n^2? solution an= 2n-1.Does that mean there is only one arithmetic sequence?
  46. C

    Indirect Proof Proof verification: sequence a_n=(−1)^n does not converge

    Theorem: Show that the sequence ## a_n = (-1)^n ## for all ## n \in \mathbb{N}, ## does not converge. My Proof: Suppose that there exists a limit ##L## such that ## a_n \rightarrow L ##. Specifically, for ## \epsilon = 1 ## there exists ## n_0 ## s.t. for all ## n > n_0## then ##|(-1)^n-L|<1##...
  47. M

    MHB How can we prove the convergence of recursive defined sequences?

    Hey! :giggle: a) Check the convergence of the sequence $a_n=\left (\frac{n+2000}{n-2000}\right)^n$, $n>1$. If it converges calculate the limit. b) Check the convergence of the recursive defined sequence $a_n=\frac{a_{n-1}}{a_{n-1}+2}$, $n>1$, with $a_1=1$.For a) we have $$a_n=\left...
  48. P

    RNA coding sequence for proteins

    Simplistic Example Given - DNA - RNA AATGTA codes for a protein. 1. Does the inverse ATGTAA usually/ever code for a protein? 2. Does the reciprocal RNA TTACAT usually/ever code for a protein?
  49. yucheng

    Prove the lower bound for a sequence (Buck, Advanced Calculus)

    Clearly, ##x_{n+1}>x_n \because x_n + \sqrt{x_n} > x_n## $$ \begin{align*} x_{n+1} &= x_n+ \sqrt{x_n} \\ &= x_1 + \sqrt{x_1} + \sqrt{x_2} + \cdots \sqrt{x_n} \\ &>n+1 \end{align*} $$ ##\because \sqrt{x_n}>\sqrt{x_1}=1## In fact, $$x_{n+1} > 1+ \sqrt{1} + \sqrt{2}+ \sqrt{3} + \cdots \sqrt{n}$$...
  50. potatocake

    How to prove rational sequence converges to irrational number

    I attempted to solve it $$ x = \frac {1}{4x} + 1 $$ $$⇒ x^2 -x -\frac{1}{4} = 0 $$ $$⇒ x = \frac{1±\sqrt2}{2} $$ However, I don't know the next step for the proof. Do I need a closed-form of xn+1or do I just need to set the limit of xn and use inequality to solve it? If I have to use...
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