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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
• 2007 AMC 12A Problems
  ...c, d</math>, and <math>e</math> be five consecutive terms in an arithmetic sequence, and suppose that <math>a+b+c+d+e=30</math>. Which of <math>a, b, c, d,</ma ...d with the term 824. Let <math>S</math> be the sum of all the terms in the sequence. What is the largest [[prime]] [[factor]] that always divides <math>S</math
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 AMC 12A Problems/Problem 7
  ...c, d</math>, and <math>e</math> be five consecutive terms in an arithmetic sequence, and suppose that <math>a+b+c+d+e=30</math>. Which of <math>a, b, c, d,</ma
  818 bytes (152 words) - 16:40, 5 April 2024
• 2007 AMC 12A Problems/Problem 16
  We can find the number of increasing [[arithmetic sequence]]s of length 3 possible from 0 to 9, and then find all the possible permuta
  2 KB (336 words) - 05:01, 4 November 2022
• 2005 AMC 12A Problems/Problem 11
  Thus, the three digits form an [[arithmetic sequence]].
  2 KB (266 words) - 00:59, 19 October 2020
• 2005 AMC 12A Problems/Problem 13
  arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?
  3 KB (430 words) - 18:52, 11 July 2020
• 2006 AIME II Problems/Problem 13
  ...h>2n+1 + 2(j-1) = 2(n+j) - 1</math>. The odd integers form an [[arithmetic sequence]] with sum <math>N = j\left(\frac{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)< ...he <math>q</math>th positive odd number, and the largest odd number in the sequence be the <math>p</math>th positive odd number. Therefore, the sum is <math>p^
  4 KB (675 words) - 10:40, 14 July 2022
• 1999 AHSME Problems
  Define a sequence of real numbers <math>a_1, a_2, a_3, \ldots</math> by <math>a_1 = 1</math> The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99
  13 KB (1,945 words) - 18:28, 19 June 2023
• Aleph null
  ...erms in any [[arithmetic sequence]], [[geometric sequence]], or [[harmonic sequence]]. It is less than, for example, aleph 1 (<math>\aleph_{1}</math>), which i
  847 bytes (120 words) - 20:49, 26 October 2007
• 2004 AMC 12A Problems/Problem 14
  A [[sequence]] of three real numbers forms an [[arithmetic progression]] with a first te ...is <math>9</math>, <math>9+d</math>, and <math>9+2d</math>. The geometric sequence (when expressed in terms of <math>d</math>) has the terms <math>9</math>, <
  4 KB (689 words) - 03:35, 16 January 2023
• 1990 USAMO Problems/Problem 4
  Let a <math>k</math>-good sequence be a sequence of distinct integers <math>\{ a_i \}_{i=1}^k</math> such that for all integ ...ood sequence, then <math>\{ a_i \}_{i=1}^k</math> is a <math>k</math>-good sequence which starts on <math>a</math>, so it is a permutation of <math>k</math> co
  3 KB (529 words) - 19:15, 18 July 2016
• 1959 AHSME Problems
  A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let S be the sum of the first nine terms of the sequence <math>x+a, x^2+2a, x^3+3a, \cdots.</math>
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  In a given arithmetic sequence the first term is <math>2</math>, the last term is <math>29</math>, and the ...>s_2</math> be the sum of the first <math>n</math> terms of the arithmetic sequence <math>17,19,\cdots</math>. Assume <math>n \ne 0</math>. Then <math>s_1=s_2<
  19 KB (3,159 words) - 22:10, 11 March 2024
• 2002 AMC 12B Problems/Problem 5
  ...ath> and <math>v, w, x, y, </math> and <math>z</math> form an [[arithmetic sequence]]. Find the value of <math>x</math>. ...etic sequence with an odd number of terms, it is simply the average of the sequence.
  2 KB (263 words) - 05:58, 8 August 2023
• 2002 AMC 12B Problems/Problem 9
  ...an increasing arithmetic sequence and <math>a,b,d</math> form a geometric sequence, then <math>\frac ad</math> is As <math>a, b, d</math> is a geometric sequence, let <math>b=ka</math> and <math>d=k^2a</math> for some <math>k>0</math>.
  2 KB (288 words) - 21:42, 11 December 2017
• 2002 AMC 12B Problems/Problem 10
  The set is an arithmetic sequence of numbers each <math>1</math> more than a multiple of <math>3</math>. Thus
  1 KB (166 words) - 00:43, 17 January 2021
• 1995 USAMO Problems
  ...n \,</math> is the least positive integer that does not form an arithmetic sequence of length <math>\, p \,</math> with any of the preceding terms. Prove that, ...itive rational number <math>\, q, \,</math> show that pressing some finite sequence of buttons will yield <math>\, q</math>. Assume that the calculator does
  3 KB (540 words) - 13:31, 4 July 2013
• Partition (combinatorics)
  ...bers is through [[generating function]]s. The generating function for the sequence <math>\{P(n)\}_{n \geq 0}</math> is given by <math>F(x)= \sum_{n \geq 0}P(n Using the formula for the sum of an [[infinite]] [[geometric sequence]] we can express this in the more compact form
  10 KB (1,508 words) - 14:24, 17 September 2017
• 2008 AMC 12A Problems
  ...s of an arithmetic sequence, and the <math>12^\text{th}</math> term of the sequence is <math>\log(b^n)</math>. What is <math>n</math>? Let <math>a_1,a_2,\ldots</math> be a sequence determined by the rule <math>a_n=a_{n-1}/2</math> if <math>a_{n-1}</math> i
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 16
  ...an [[arithmetic sequence]], and the <math>12^\text{th}</math> term of the sequence is <math>\log{b^n}</math>. What is <math>n</math>? The first three terms of the arithmetic sequence are <math>3A + 7B</math>, <math>5A + 12B</math>, and <math>8A + 15B</math>,
  3 KB (577 words) - 16:33, 9 October 2022
• 2008 AMC 12B Problems/Problem 24
  ...tween the x coordinates of consecutive <math>A_i</math> form an arithmetic sequence (<math>x_{A_1} - x_{A_0} = \frac{2}{3}</math>, <math>x_{A_2} - x_{A_1} = \f
  9 KB (1,482 words) - 13:52, 4 April 2024
• Symmetric group
  ...esentations of permutations: as functions, as products of [[cycle]]s, as [[sequence]]s or [[word]]s, etc.) Knowledge of the symmetric group <math>S_{n}</math>
  10 KB (1,668 words) - 15:33, 25 May 2008
• 2008 AIME II Problems/Problem 1
  Rewriting this sequence with more terms, we have ...(a+1)^2 - a^2 = 8a + 12</cmath>, turning <math>N</math> into a arithmetic sequence with 25 terms, them being <math>1, 5, 9, \dots ,97</math>, as the series <m
  4 KB (575 words) - 16:41, 14 April 2024
• 2008 Mock ARML 1 Problems
  ...g it with one of the opposite color. Compute the probability that, after a sequence of turns, there are <math>5</math> black balls in the hat before there are
  3 KB (409 words) - 16:41, 29 May 2008
• 2008 iTest Problems
  ...st <math>1</math> that he rolled. His first <math>31</math> rolls make the sequence <math>4,3,11,3,11,8,5,2,12,9,5,7,11,3,6,10,\textbf{1},8,3,\textbf{2},10,4,2 ...ng - do <math>\textit{not}</math> assume he starts by rolling the specific sequence of <math>31</math> rolls above.)
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2008 iTest Problems/Problem 94
  sequence ...7</math> and <math>2008</math> must be a divisor of some term in the given sequence. The largest prime less than <math>2008</math> is <math>2003</math>, which
  4 KB (571 words) - 21:21, 22 November 2018
• Prime triplet
  A set of three [[prime number]]s which form an arithmetic sequence with common difference two is called a '''prime triplet'''.
  836 bytes (121 words) - 00:59, 17 March 2009
• 1973 USAMO Problems/Problem 5
  ...t the cube roots of three distinct prime numbers cannot form an arithmetic sequence.
  4 KB (683 words) - 20:18, 29 December 2019
• 2002 AMC 10B Problems
  Suppose that <math>\{a_n\}</math> is an arithmetic sequence with Let <math>\{a_k\}</math> be a sequence of integers such that <math>a_1=1</math> and <math>a_{m+n}=a_m+a_n+mn,</mat
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2004 AMC 10B 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 (1,988 words) - 23:06, 7 March 2024
• 2000 AMC 10 Problems
  ...is the last to appear in the units position of a number in the Fibonacci sequence?
  14 KB (2,035 words) - 21:57, 2 May 2024
• 1999 AHSME Problems/Problem 20
  The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99
  2 KB (268 words) - 14:00, 21 March 2023
• 1985 AJHSME Problems/Problem 2
  ...2}\cdot(a_1+a_n)</math> where <math>n</math> is the number of terms in the sequence, <math>a_1</math> is the first term and <math>a_n</math> is the last term.
  2 KB (282 words) - 13:43, 4 April 2024
• 1986 AJHSME Problems/Problem 25
  ...the largest (since the sum of the 2 entries is twice the average of whole sequence). <math>2+100=102</math>, <math>3+99=102</math>, <math>4+100=104</math>, <m
  2 KB (291 words) - 20:13, 17 January 2024
• 2004 AMC 10B Problems/Problem 21
  ...t <math>S</math> is the union of the first <math>2004</math> terms of each sequence. How many distinct numbers are in <math>S</math>?
  2 KB (357 words) - 16:20, 5 May 2024
• 2009 AMC 12A Problems
  ...h>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence is <math>2009</math>. What is <math>n</math>? ..._2</math>, <math>F_3</math>, and <math>F_4</math> shown are the first in a sequence of figures. For <math>n\ge3</math>, <math>F_n</math> is constructed from <m
  13 KB (2,105 words) - 13:13, 12 August 2020
• 2009 AMC 12A Problems/Problem 7
  ...h>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence is <math>2009</math>. What is <math>n</math>? As this is an arithmetic sequence, the difference must be constant: <math>(5x-11) - (2x-3) = (3x+1) - (5x-11)
  825 bytes (128 words) - 10:17, 9 February 2015
• 1988 AJHSME Problems
  What is the <math>100\text{th}</math> number in the arithmetic sequence: <math>1,5,9,13,17,21,25,...</math>?
  14 KB (1,872 words) - 15:23, 17 January 2023
• 2009 AIME I Problems/Problem 7
  The sequence <math>(a_n)</math> satisfies <math>a_1 = 1</math> and <math>5^{(a_{n + 1} - Plug in <math>n = 1, 2, 3, 4</math> to see the first few terms of the sequence: <cmath>\log_5{5},\log_5{8}, \log_5{11}, \log_5{14}.</cmath> We notice that
  2 KB (340 words) - 00:26, 9 January 2023
• 1988 AJHSME Problems/Problem 19
  What is the <math>100\text{th}</math> number in the [[arithmetic sequence]]: <math>1,5,9,13,17,21,25,...</math>? To get from the <math>1^\text{st}</math> term of an arithmetic sequence to the <math>100^\text{th}</math> term, we must add the common [[difference
  946 bytes (133 words) - 10:51, 28 June 2023
• 2009 IMO Problems/Problem 3
  Suppose that <math>s_1,s_2,s_3,\ldots</math> is a strictly increasing sequence of positive integers such that the subsequences are both arithmetic progressions. Prove that the sequence <math>s_1,s_2,s_3,\ldots</math> is itself an arithmetic progression.
  1 KB (184 words) - 01:16, 19 November 2023
• 2009 IMO Problems
  Suppose that <math>s_1,s_2,s_3,\ldots</math> is a strictly increasing sequence of positive integers such that the subsequences are both arithmetic progressions. Prove that the sequence <math>s_1,s_2,s_3,\ldots</math> is itself an arithmetic progression.
  3 KB (509 words) - 09:23, 10 September 2020
• 1995 USAMO Problems/Problem 1
  ...n \,</math> is the least positive integer that does not form an arithmetic sequence of length <math>\, p \,</math> with any of the preceding terms. Prove that, ...n that <math>a_{n+1}>a_{n}</math> (without this assumption, I can have the sequence
  4 KB (625 words) - 18:23, 22 March 2024
• Unique factorization domain
  So now we can construct a sequence <math>r_1,r_2,\ldots</math> of elements of <math>S</math> such that
  6 KB (1,217 words) - 23:05, 23 August 2009
• 2010 AMC 12A Problems
  ...nd <math>3p+q</math>. What is the <math>2010^\text{th}</math> term of this sequence?
  12 KB (1,817 words) - 15:00, 12 August 2020
• 2010 AMC 12A Problems/Problem 20
  ...integer terms with <math>a_1=b_1=1</math>, we can write the terms of each sequence as ...th>(m-1)</math> times the common difference for that particular arithmetic sequence. Let the common difference of <math>(a_n)</math> be <math>k</math> and the
  5 KB (797 words) - 15:27, 3 July 2023
• 2010 AMC 12A Problems/Problem 10
  ...nd <math>3p+q</math>. What is the <math>2010^\text{th}</math> term of this sequence?
  1 KB (178 words) - 20:47, 27 October 2022
• 2010 AMC 12A Problems/Problem 23
  ...liminated. After factoring out a 2 from each of the 9 even numbers in this sequence, the 10, 20, 30, ..., 90 becomes 1, 2, 3, 4, 1, 6, 7, 8, 9, whose product i
  10 KB (1,525 words) - 09:44, 24 April 2024
• 2010 AMC 10B Problems
  ...the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neithe
  12 KB (1,817 words) - 22:44, 22 December 2020
• 2010 AMC 12B Problems
  ...the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neithe A geometric sequence <math>(a_n)</math> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, an
  12 KB (1,845 words) - 13:00, 19 February 2020
• 2010 USAJMO Problems/Problem 1
  ...ch element of <math>[n]</math> appears precisely one time as a term of the sequence. For example, <math>(3, 5, 1, 2, 4)</math> is a permutation of <math>[5]</m ...his is acceptable, as <math>ka_k</math> is always <math>k^2</math> in this sequence.
  12 KB (2,338 words) - 20:30, 13 February 2024

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