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• Arithmetic sequence
  ...ms is constant. This constant is called the '''common difference''' of the sequence. ...difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with common difference <math>-8</math>; however, <math>7, 0, 7, 14</math> a
  4 KB (736 words) - 02:00, 7 March 2024
• Arithmetic Sequence
  #REDIRECT[[Arithmetic sequence]]
  32 bytes (3 words) - 17:05, 3 September 2021

Page text matches

• 2016 AMC 10A Problems/Problem 10
  ..., 2x+14</math> is an arithmetic progression, meaning that the terms in the sequence increase by the same amount each term.
  2 KB (337 words) - 14:56, 25 June 2023
• Euler's Totient Theorem
  ...MC 10B]]) Let <math>a_1,a_2,\dots,a_{2018}</math> be a strictly increasing sequence of positive integers such that <cmath>a_1+a_2+\cdots+a_{2018}=2018^{2018}.<
  3 KB (542 words) - 17:45, 21 March 2023
• Algebra/Intermediate
  * [[Sequence | Sequences]] and [[Series]] ** [[Arithmetic sequence]]
  2 KB (198 words) - 17:47, 3 November 2021
• Prime number
  3,5 and 7 are the only primes that form an [[Arithmetic sequence]] with common difference 2. 3,7 and 11 are the only primes that form an [[Arithmetic sequence]] with common difference 4.
  6 KB (985 words) - 12:38, 25 February 2024
• Muirhead's Inequality
  ...y''' states that if a sequence <math>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n The inequality is easier to understand given an example. Since the sequence <math>(5,1)</math> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</mat
  8 KB (1,346 words) - 12:53, 8 October 2023
• Logarithm
  * The [[sequence]] <math> a_1, a_2, \ldots </math> is [[geometric sequence|geometric]] with <math> a_1=a </math> and common [[ratio]] <math> r, </math
  4 KB (680 words) - 12:54, 16 October 2023
• Sequence
  A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[in ...th>f(x) = x^2</math> defined on <math>\mathbb{N}</math> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldots)</math>.
  2 KB (413 words) - 21:18, 13 November 2022
• Geometric sequence
  ...e terms is constant. This constant is called the '''common ratio''' of the sequence. ...n ratio <math>2</math> and <math>100, -50, 25, -25/2</math> is a geometric sequence with common ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and
  4 KB (644 words) - 12:55, 7 March 2022
• Arithmetic sequence
  ...ms is constant. This constant is called the '''common difference''' of the sequence. ...difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with common difference <math>-8</math>; however, <math>7, 0, 7, 14</math> a
  4 KB (736 words) - 02:00, 7 March 2024
• Geometry
  ...s length, width, and position. It is two-dimensional. The point/line/plane sequence can be extended to spaces and higher dimensions.
  3 KB (393 words) - 07:59, 25 September 2020
• Limit
  ...h> has a limit <math>L = \lim_{x \rightarrow c} f(x)</math> if for every [[sequence]] <math>\left\langle x_n \right\rangle</math> that converges to <math>c</ma
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2006 AIME I Problems/Problem 15
  Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all ..._{k + 1}|\pm3 = |x_k|\pm3\mp3 = |x_k| = |x_{k - 1} + 3|</math>. So the new sequence works under the same criteria as the old one. In this way, we can pair all
  6 KB (910 words) - 19:31, 24 October 2023
• 2021 JMC 10
  ...bers <cmath>24_b,n,57_b,72_b, \ldots</cmath> form an increasing arithmetic sequence in that specific order. Then, what is the value of <math>n,</math> expresse ...1</math> for positive integers <math>n \ge1.</math> How many terms of this sequence are divisible by <math>99?</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2005 AMC 12A Problems
  arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  A sequence of three real numbers forms an arithmetic progression with a first term of
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2000 AMC 12 Problems
  ...s is the last to appear in the units position of a number in the Fibonacci sequence?
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2002 AMC 12B Problems
  .../math> and <math>v, w, x, y, </math> and <math>z</math> form an arithmetic sequence. Find the value of <math>x</math>. ...an increasing arithmetic sequence and <math>a,b,d</math> form a geometric sequence, then <math>\frac ad</math> is
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2004 AMC 12B Problems
  In the sequence <math>2001</math>, <math>2002</math>, <math>2003</math>, <math>\ldots</math <math>2004^\textrm{th}</math> term in this sequence?
  13 KB (2,049 words) - 13:03, 19 February 2020
• Set
  .... If <math>AB, BC, CD, DE,</math> and <math>EA</math> form an [[arithmetic sequence]] (not necessarily in increasing order), find the value of <math>CD</math>.
  11 KB (2,021 words) - 00:00, 17 July 2011
• 1991 AJHSME Problems
  ...lying on a table, the paper is folded in half four times in the following sequence:
  17 KB (2,246 words) - 13:37, 19 February 2020
• 2022 USAJMO Problems/Problem 1
  ...sequence of integers <math>a_1,a_2,\cdots</math> and an infinite geometric sequence of integers <math>g_1,g_2,\cdots</math> satisfying the following properties ...metic sequence be <math>\{ a, a+d, a+2d, \dots \}</math> and the geometric sequence to be <math>\{ g, gr, gr^2, \dots \}</math>. Rewriting the problem based on
  4 KB (792 words) - 00:29, 13 April 2024
• 2005 AIME I Problems
  ...mon difference is <math> k. </math> For example, <math> S_3 </math> is the sequence <math> 1,4,7,10,\ldots. </math> For how many values of <math> k </math> doe
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 2
  ...n difference is <math> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <math> k </math> d Suppose that the <math>n</math>th term of the sequence <math>S_k</math> is <math>2005</math>. Then <math>1+(n-1)k=2005</math> so <
  2 KB (303 words) - 01:31, 5 December 2022
• 2004 AIME I Problems/Problem 2
  ...nd last terms of <math>A</math>. This comes from the sum of an arithmetic sequence. ...lso note how exactly i used the fact that the first and last terms of each sequence sum to <math>4</math> and <math>1</math> respectively (add <math>x</math> a
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME II Problems/Problem 9
  ...rithmetic progression. Let <math> a_n </math> be the greatest term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...cdot 33 = 957</math>, and this is the <math>2(8) = 16</math>th term of the sequence.
  3 KB (538 words) - 21:33, 30 December 2023
• 2004 AIME II Problems
  ...rithmetic progression. Let <math> a_n </math> be the greatest term in this sequence that is less than 1000. Find <math> n+a_n. </math>
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1999 AIME Problems
  Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  A sequence of numbers <math>x_{1},x_{2},x_{3},\ldots,x_{100}</math> has the property t
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2002 AIME I Problems
  Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each th Consider the sequence defined by <math>a_k=\frac 1{k^2+k}</math> for <math>k\ge 1</math>. Given t
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  In an increasing sequence of four positive integers, the first three terms form an arithmetic progres
  6 KB (965 words) - 16:36, 8 September 2019
• 2003 AIME II Problems
  ...B</math>, and <math>C</math> - some of these letters may not appear in the sequence - and in which <math>A</math> is never immediately followed by <math>B</mat Find the eighth term of the sequence <math>1440,</math> <math>1716,</math> <math>1848,\ldots,</math> whose terms
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1984 AIME Problems/Problem 1
  Since we are dealing with an arithmetic sequence,
  4 KB (576 words) - 21:03, 23 December 2023
• 1989 AIME Problems/Problem 8
  ...tic sequences must be constant (but nonzero). One example is the following sequence of perfect squares: Let <math>s_n = n^2</math> be the sequence of perfect squares.
  8 KB (1,146 words) - 04:15, 20 November 2023
• 1999 AIME Problems/Problem 1
  Find the smallest prime that is the fifth term of an increasing [[arithmetic sequence]], all four preceding terms also being [[prime number|prime]]. ...ind that <math>5,11,17,23</math>, and <math>29</math> form an [[arithmetic sequence]]. Thus, the answer is <math>029</math>.
  2 KB (332 words) - 13:22, 3 August 2020
• 2000 AIME I Problems/Problem 15
  ...<math>2000</math> is a small number. If you don't want to do this, define sequence <math>a_n = 2a_{n-1} - 1</math>, and solve for the closed form, which is ve
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 10
  A [[sequence]] of numbers <math>x_{1},x_{2},x_{3},\ldots,x_{100}</math> has the property Let the sum of all of the terms in the sequence be <math>\mathbb{S}</math>. Then for each integer <math>k</math>, <math>x_k
  2 KB (319 words) - 22:26, 29 December 2022
• 2003 AIME I Problems/Problem 8
  In an [[increasing sequence]] of four positive integers, the first three terms form an [[arithmetic pro The sequence is of the form <math>a-d,</math> <math>a,</math> <math>a+d,</math> <math>\f
  5 KB (921 words) - 23:21, 22 January 2023
• 2003 AIME II Problems/Problem 8
  Find the eighth term of the sequence <math>1440,</math> <math>1716,</math> <math>1848,\ldots,</math> whose terms Let the first sequence be
  5 KB (793 words) - 15:18, 14 July 2023
• 2000 AMC 12 Problems/Problem 8
  ...ferences are constant and all equal to <math>4</math>. Thus, the original sequence can be generated from a quadratic function. ...erm being <math>4</math> and the difference being <math>4</math>. Let this sequence be <math>a_n</math>
  7 KB (988 words) - 15:14, 10 April 2024
• Arithmetico-geometric series
  ...ometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined as: <math>x_n=a_ng_n</math>, where <math>a_n</math> and <math>g_n</ ...f the first <math>n</math> terms of an <math>\textbf{arithmetico-geometric sequence}</math> is <math>\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2
  2 KB (477 words) - 19:39, 17 August 2020
• Series
  A '''series''' is a sum of consecutive terms in a [[sequence]]. Common series are based on common sequences. * [[Sequence]]
  400 bytes (43 words) - 21:21, 22 July 2021
• 2005 AMC 10A Problems/Problem 17
  ...h not necessarily in that order. What is the middle term of the arithmetic sequence? ...ddle term in an arithmetic sequence is the average of all the terms in the sequence, the middle number is <math>\frac{60}{5}=\boxed{\textbf{(D) }12}</math>
  2 KB (266 words) - 03:36, 16 January 2023
• Arithmetic progression
  #REDIRECT[[Arithmetic sequence]]
  32 bytes (3 words) - 23:09, 25 December 2021
• Arithmetic series
  #REDIRECT[[Arithmetic sequence]]
  32 bytes (3 words) - 11:29, 31 August 2021
• 2007 iTest Problems
  If every possible sequence of scores is equally likely, what is the expected score of the losing team? ...>f(2) = 2</math> because the <math>100^{\text{th}}</math> digit enters the sequence in the placement of the two-digit integer <math>55</math>. Find the value o
  30 KB (4,794 words) - 23:00, 8 May 2024
• 2005 AMC 10A Problems
  ...ce, although not necessarily in this order. What is the middle term of the sequence?
  14 KB (2,026 words) - 11:45, 12 July 2021
• 2004 AMC 10A Problems
  A sequence of three real numbers form an arithmetic progression with a first term of 9
  15 KB (2,092 words) - 20:32, 15 April 2024
• 2020 AMC 10A Problems
  ...4), (4,4),</math> and <math>(4,0)</math>. What is the probability that the sequence of jumps ends on a vertical side of the square? ...How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are n
  13 KB (1,968 words) - 18:32, 29 February 2024
• 2007 AIME I Problems/Problem 7
  ...ula for the sum of an [[arithmetic sequence]] and the sum of a [[geometric sequence]] yields that our answer is <math>\left[\frac{(1000 + 1)(1000)}{2} - (1 + 2
  2 KB (242 words) - 20:26, 20 April 2023
• 2007 AIME II Problems/Problem 12
  The increasing [[geometric sequence]] <math>x_{0},x_{1},x_{2},\ldots</math> consists entirely of [[integer|inte ...<math>a^8r^{28} = 3^{308}</math>. Since all of the terms of the geometric sequence are integral powers of <math>3</math>, we know that both <math>a</math> and
  5 KB (829 words) - 12:22, 8 January 2024

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