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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.
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• 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.
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• 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?
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• 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.
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• 2010 USAJMO Problems/Problem 2
  The sequence is <math>2, 4, 6, \ldots, 2n-2</math>. We will prove that any sequence <math>x_1, \ldots, x_{n-1}</math>, that satisfies
  6 KB (1,177 words) - 23:09, 20 November 2022
• 2010 AMC 12B Problems/Problem 19
  ...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 ...e quarterly scores for the Raiders. We know <math>r > 1</math> because the sequence is said to be increasing. We also know that each of <math>a, ar, ar^2, ar^3
  6 KB (990 words) - 19:14, 25 March 2024
• 2011 AMC 10A Problems
  In the eight-term sequence <math>A,B,C,D,E,F,G,H</math>, the value of <math>C</math> is 5 and the sum
  13 KB (1,903 words) - 18:09, 19 April 2021
• 1993 AHSME Problems
  ...<math>Q=R_{24}/R_4</math> is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in <math>Q</math> is: Consider the non-decreasing sequence of positive integers
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2011 AMC 10A Problems/Problem 4
  We see that both sequences have equal numbers of terms, so reformat the sequence to look like: ...y corresponding number in X's sequence. Since there are 46 numbers in each sequence, the difference must be
  2 KB (323 words) - 12:07, 19 October 2023
• 2011 AMC 10A Problems/Problem 23
  ...ndice: } 14,41,68,\cdots,986.</cmath>Notice how each list is an arithmetic sequence where the common differene is thrice the common ratio of the previous list We apply the function <math>f(x)=3x-1</math> six times to attain the sequence of numbers <math>1,2,5,14,41,122,\boxed{\textbf{(C)}\;365}</math>.
  6 KB (926 words) - 23:38, 8 April 2024
• 2007 AMC 8 Problems/Problem 16
  ...h>\pi</math>. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much m
  2 KB (332 words) - 12:22, 16 August 2021
• 2011 AMC 12B Problems
  ...CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>?
  13 KB (1,978 words) - 16:28, 12 July 2020
• 2001 AMC 10 Problems/Problem 22
  ...t any <math>3</math> numbers that go through the middle form an arithmetic sequence. ...for x in this system we get <math>x=22</math>, so now using the arithmetic sequence knowledge we find that <math>y=26</math> and <math>z=20</math>.
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• 2011 AIME II Problems/Problem 3
  ...[[convex polygon|convex]] 18-sided polygon form an increasing [[arithmetic sequence]] with integer values. Find the degree measure of the smallest [[angle]]. ...60+d)^\circ</math>. Since the step is <math>2d</math> the last term of the sequence is <math>(160 + 17d)^\circ</math>, which must be less than <math>180^\circ<
  3 KB (547 words) - 01:13, 31 January 2024
• 2011 AIME II Problems
  ...s of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. The sum of the first <math>2011</math> terms of a geometric sequence is <math>200</math>. The sum of the first <math>4022</math> terms is <math>
  8 KB (1,301 words) - 08:43, 11 October 2020
• 1992 AJHSME Problems/Problem 25
  ...ount the number of terms or pourings, as the numerators form an arithmetic sequence with a common difference of 1 and endpoints (1,9), the number of pourings i
  1 KB (175 words) - 22:45, 25 May 2021
• 2003 AMC 10B Problems
  The second and fourth terms of a geometric sequence are <math>2</math> and <math>6</math>. Which of the following is a possible The first four terms in an arithmetic sequence are <math>x + y, x - y, xy,</math> and <math>x/y,</math> in that order. Wha
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• 2002 AMC 10B Problems/Problem 19
  Suppose that <math>\{a_n\}</math> is an arithmetic sequence with
  3 KB (472 words) - 14:56, 17 August 2023
• 1984 AHSME Problems/Problem 12
  If the [[sequence]] <math> \{a_n\} </math> is defined by We begin to evaluate the first couple of terms of the sequence, hoping to find a pattern:
  2 KB (325 words) - 18:10, 30 November 2013
• 2003 AMC 10B Problems/Problem 24
  The first four terms in an arithmetic sequence are <math>x+y</math>, <math>x-y</math>, <math>xy</math>, and <math>\frac{x} Because this is an arithmetic sequence, we conclude from the first two terms that the common difference is <math>-
  4 KB (779 words) - 16:16, 12 March 2024
• 1980 USAMO Problems
  ...aximum possible number of three term arithmetic progressions in a monotone sequence of <math>n</math> distinct reals.
  2 KB (276 words) - 13:41, 26 December 2015
• 2012 AMC 10A Problems/Problem 10
  ...sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? ..., a+2r, \cdots. a+11r</math>. Use the formula for the sum of an arithmetic sequence and set it equal to 360, the number of degrees in a circle.
  4 KB (592 words) - 13:54, 1 July 2023
• 2012 AMC 10A Problems
  ...sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
  13 KB (1,994 words) - 01:31, 22 December 2023
• 2012 AMC 12A Problems
  ...sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? Let <math>\{a_k\}_{k=1}^{2011}</math> be the sequence of real numbers defined by <math>a_1=0.201,</math> <math>a_2=(0.2011)^{a_1}
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2012 AIME I Problems/Problem 2
  ...</math>. Find the sum of the first, last, and middle terms of the original sequence. ...6 = n^2 + 715 \rightarrow n=11.</math> Now the middle term of the original sequence is simply the average of all the terms, or <math>\frac{715}{11} = 65,</math
  3 KB (451 words) - 18:30, 20 January 2024
• 2015 AIME I Problems
  ...1,a_2,a_3 \le 10</math>. Each ordered triple in <math>S</math> generates a sequence according to the rule <math>a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |</math> for
  10 KB (1,615 words) - 21:48, 13 January 2024
• 2012 AIME I Problems
  ...6</math>. Find the sum of the first, last, and middle term of the original sequence. ...(x, y)</math> with <math>|x| + |y| \le 100</math> that can be reached by a sequence of such jumps. Find the remainder when <math>M</math> is divided by <math>1
  10 KB (1,617 words) - 14:49, 2 June 2023
• Mock AIME II 2012 Problems
  ...a_1, a_2+1, a_3+8, a_4+27, a_5+64, a_6+125, \cdots</math> is an arithmetic sequence, find the smallest positive integer value of <math>x</math> such that <math
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• Mock AIME II 2012 Problems/Problem 15
  ...a_1, a_2+1, a_3+8, a_4+27, a_5+64, a_6+125, \cdots</math> is an arithmetic sequence, find the smallest positive integer value of <math>x</math> such that <math ...e <math>a_1, a_2+1, a_3+8, a_4+27, a_5+64, a_6+125</math> is an arithmetic sequence, difference(let this be h)<math>=a_2-a_1+1=a_3-a_2+7=a_4-a_3+19=a_5-a_4+37=
  4 KB (660 words) - 15:55, 8 March 2015
• 1988 USAMO Problems/Problem 5
  By a limiting process, we can extend the problem to that of finding a sequence <math>b_1, b_2, \ldots</math> of integers such that
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• 1966 AHSME Problems/Problem 18
  In a given arithmetic sequence the first term is <math>2</math>, the last term is <math>29</math>, and the
  492 bytes (69 words) - 03:33, 15 February 2019
• 1955 AHSME Problems
  ...first term <math>\neq 0</math> and <math>r \neq 0</math> and an arithmetic sequence with the first term <math>=0</math>. A third sequence <math>1,1,2\ldots</math> is formed by adding corresponding terms of the two
  22 KB (3,509 words) - 21:29, 31 December 2023
• 1980 AHSME Problems
  In the non-decreasing sequence of odd integers <math>\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}
  15 KB (2,302 words) - 10:47, 30 April 2021
• 1980 USAMO Problems/Problem 2
  ...aximum possible number of three term arithmetic progressions in a monotone sequence of <math>n</math> distinct reals. ...math>n</math> with the entire <math>n</math> numbers forming an arithmetic sequence <cmath>(1, 2, 3, \ldots, n)</cmath>
  2 KB (405 words) - 19:04, 9 August 2023
• 2005 AMC 8 Problems/Problem 12
  ...and <math>n-4(6)=n-24</math> bananas on May 1. The sum of this arithmetic sequence is equal to <math>100</math>. Simply realize that the middle term of the arithmetic sequence is the arithmetic mean of all terms, which is simply <math>\frac{100}{5}=20
  2 KB (293 words) - 11:48, 4 May 2022
• 1993 AHSME Problems/Problem 21
  Let <math>a_1,a_2,\cdots,a_k</math> be a finite arithmetic sequence with <math>a_4 +a_7+a_{10} = 17</math> and <math>a_4+a_5+\cdots+a_{13} +a_{
  1,013 bytes (161 words) - 17:49, 26 August 2017
• 2013 AMC 12A Problems
  The sequence <math>S_1, S_2, S_3, \cdots, S_{10}</math> has the property that every term The sequence
  14 KB (2,204 words) - 20:25, 22 November 2020
• 2013 AMC 10A Problems/Problem 4
  ...</math> when the team has even runs. The opponents will have an arithmetic sequence of even runs, <math>2,4,6,8,10</math>, when the team has odd runs. The sum
  2 KB (303 words) - 21:40, 4 July 2023
• 2013 AMC 12A Problems/Problem 14
  The sequence Since the sequence is arithmetic,
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• 2013 AMC 12A Problems/Problem 20
  2. the number of ways to choose <math>x</math> or start the sequence, which is <math>19-(a+b)</math>
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• 2013 AMC 10B Problems/Problem 19
  The real numbers <math>c,b,a</math> form an arithmetic sequence with <math>a \geq b \geq c \geq 0</math>. The quadratic <math>ax^2+bx+c</ma ...ot change the roots or the fact that the coefficients are in an arithmetic sequence. Also, we know that there is exactly one root so this equation must be of t
  5 KB (969 words) - 19:14, 15 August 2023
• 2013 AMC 12B Problems
  ...e third is the sum of the previous two terms, and the seventh term of each sequence is <math>N</math>. What is the smallest possible value of <math>N</math> ?
  16 KB (2,459 words) - 02:46, 30 January 2021
• 2013 AMC 12B Problems/Problem 13
  Since the angles of Quadrilateral <math>ABCD</math> form an arithmetic sequence, we can assign each angle with the value <math>a</math>, <math>a+d</math>, .... In this case, <math>\alpha, \beta, \gamma</math> also form an arithmetic sequence with 45, 60, and 75, and the largest two angles of the quadrilateral add up
  3 KB (443 words) - 12:32, 8 January 2021
• 2013 AMC 10B Problems
  The real numbers <math>c,b,a</math> form an arithmetic sequence with <math>a\ge b\ge c\ge 0</math>. The quadratic <math>ax^2+bx+c</math> ha ...third, is the sum of the previous two terms, and the seventh term of each sequence is <math>N</math>. What is the smallest possible value of <math>N</math>?
  12 KB (1,926 words) - 21:54, 6 October 2022
• 1991 AHSME Problems/Problem 12
  ...(in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of integers. Let <math>m</math> be the measure of the largest interior angl
  1 KB (195 words) - 17:35, 23 February 2018
• 1991 AHSME Problems
  ...(in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of integers. Let <math>m</math> be the measure of the largest interior angl ...math>N</math> is a ''palindrome'' if the integer obtained by reversing the sequence of digits of <math>N</math> is equal to <math>N</math>. The year 1991 is th
  16 KB (2,451 words) - 04:27, 6 September 2021
• Mock AIME I 2015 Problems
  Let <math>a,b,c</math> be consecutive terms (in that order) in an arithmetic sequence with common difference <math>d</math>. Suppose <math>\cos b</math> and <ma
  9 KB (1,463 words) - 14:48, 12 February 2017
• 1966 AHSME Problems/Problem 19
  ...>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<
  659 bytes (111 words) - 02:30, 15 September 2014
• 1962 AHSME Problems/Problem 32
  ...m. Letting <math>a=1</math> and <math>r=\frac12</math>, we can rewrite the sequence as <math>a, a+r, \dots, a+(n-1)r</math>. Recall that the sum of the first <math>n</math> terms of an arithmetic sequence is <math>na+\binom{n}2r</math>.
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• 1994 AHSME Problems
  In the sequence ...ath> and <math>x \neq y</math>. If <math>x, 2y, 3z</math> is an arithmetic sequence, then <math>r</math> is
  14 KB (2,124 words) - 13:39, 19 February 2020
• 1981 AHSME Problems
  In a geometric sequence of real numbers, the sum of the first <math>2</math> terms is <math>7</math
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• 1994 AHSME Problems/Problem 20
  ...ath> and <math>x \neq y</math>. If <math>x, 2y, 3z</math> is an arithmetic sequence, then <math>r</math> is ...<math>y=xr, z=xr^2</math>. Since <math>x, 2y, 3z</math> are an arithmetic sequence, there is a common difference and we have <math>2xr-x=3xr^2-2xr</math>. Div
  824 bytes (139 words) - 03:15, 28 May 2021
• 1960 AHSME Problems
  You are given a sequence of <math>58</math> terms; each term has the form <math>P+n</math> where <ma Let <math>N</math> be the number of primes appearing in this sequence. Then <math>N</math> is:
  21 KB (3,242 words) - 21:27, 30 December 2020
• 1973 AHSME Problems
  ...<math> \log_an</math>, <math> \log_bn</math>, <math> \log_cn</math> form a sequence
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1990 AHSME Problems/Problem 3
  The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is <math>75^\circ</math>, then the largest angle is
  674 bytes (97 words) - 03:44, 4 February 2016
• 1968 AHSME Problems/Problem 20
  ...ause this particular polygon is convex and has its angles in an arithmetic sequence with its largest angle being <math>160</math>, we can find the sum of the a Plugging this into the formula for finding the sum of an arithmetic sequence...
  1 KB (182 words) - 01:53, 16 August 2023
• 1990 AHSME Problems
  The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is <math>75^\circ</math>, then the largest angle is
  14 KB (2,099 words) - 01:15, 10 September 2021
• 1969 AHSME Problems/Problem 9
  ...<math>2+51=53</math>, so the sum of the first <math>52</math> terms of the sequence is <math>\frac{52(2+53)}{2} = 1430</math>. Thus, the [[arithmetic mean]] i
  905 bytes (123 words) - 03:53, 7 June 2018
• 1969 AHSME Problems/Problem 33
  ...the first arithmetic sequence and <math>T</math> be the second arithmetic sequence. If <math>n = 1</math>, then <math>S_1:T_1 = 8:31</math>. Since <math>S_1 ...know that the 11th term is the average value of the first 21 terms in each sequence. So the desired ratio of the 11th terms is just <math>(21 \cdot 7 + 1) : (2
  3 KB (572 words) - 23:44, 3 April 2023
• 1969 AHSME Problems
  Let a sequence <math>\{u_n\}</math> be defined by <math>u_1=5</math> and the relationship
  16 KB (2,662 words) - 14:12, 20 February 2020
• 1952 AHSME Problems/Problem 30
  ...our common difference be <math>d</math>. Thus, the first few terms of the sequence are <math>a</math>, <math>a + d</math>, <math>a + 2d</math>, ...
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• 1980 AHSME Problems/Problem 11
  ...erm of the sequence and let <math>d</math> be the common difference of the sequence.
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• 1987 AHSME Problems
  The first four terms of an arithmetic sequence are <math>a, x, b, 2x</math>. The ratio of <math>a</math> to <math>b</math> Consider the sequence of numbers defined recursively by <math>t_1=1</math> and for <math>n>1</mat
  16 KB (2,291 words) - 13:45, 19 February 2020
• 1979 AHSME Problems
  ...math>n\ge 2</math>, the product of the first <math>n</math> numbers in the sequence is <math>n^2</math>. The sum of the third and the fifth numbers in the sequence is
  17 KB (2,664 words) - 01:34, 19 March 2022
• 1977 AHSME Problems
  If <math>a_1,a_2,a_3,\dots</math> is a sequence of positive numbers such that <math>a_{n+2}=a_na_{n+1}</math> for all posit then the sequence <math>a_1,a_2,a_3,\dots</math> is a geometric progression
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1976 AHSME Problems
  For a sequence <math>u_1,u_2\dots</math>, define <math>\Delta^1(u_n)=u_{n+1}-u_n</math> an
  17 KB (2,835 words) - 14:36, 8 September 2021
• 1975 AHSME Problems
  In the sequence of numbers <math>1, 3, 2, \ldots</math> each term after the first two is eq The sum of the first one hundred terms of the sequence is
  16 KB (2,512 words) - 04:48, 27 November 2021
• 1972 AHSME Problems
  The sum of the first <math>n</math> terms of the sequence <math>1,~(1+2),~(1+2+2^2),~\dots ~(1+2+2^2+\dots +2^{n-1})</math> in terms
  17 KB (2,732 words) - 13:54, 20 February 2020
• 2007 iTest Problems/Problem 29
  The list of numbers is an [[arithmetic sequence]] with <math>2007</math> terms, first term <math>1</math>, and last term <m
  601 bytes (83 words) - 19:16, 10 June 2018
• 1965 AHSME Problems
  Given the sequence <math>10^{\frac {1}{11}},10^{\frac {2}{11}},10^{\frac {3}{11}},\ldots,10^{\ ...lue of n such that the product of the first <math>n</math> members of this sequence exceeds <math>100000</math> is:
  19 KB (2,873 words) - 18:57, 16 August 2023
• 1957 AHSME Problems
  ...largest of the following integers which divides each of the numbers of the sequence <math>1^5 - 1,\, 2^5 - 2,\, 3^5 - 3,\, \cdots, n^5 - n, \cdots</math> is:
  26 KB (3,950 words) - 21:09, 31 August 2020
• 1987 AHSME Problems/Problem 9
  The first four terms of an arithmetic sequence are <math>a, x, b, 2x</math>. The ratio of <math>a</math> to <math>b</math> ...h>, we have <math>2d = 2x -x = x \implies d = \frac{x}{2}</math>. Thus the sequence is <math>\frac{x}{2}, x, \frac{3x}{2}, 2x</math>, so the ratio is <math>\fr
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• 2015 AMC 10A Problems
  How many terms are there in the arithmetic sequence <math>13</math>, <math>16</math>, <math>19</math>, . . ., <math>70</math>,
  12 KB (1,897 words) - 22:45, 18 March 2024
• 2015 AMC 10A Problems/Problem 7
  How many terms are in the arithmetic sequence <math>13</math>, <math>16</math>, <math>19</math>, <math>\dotsc</math>, <ma ...math>\dotsc</math>, <math>70</math>, <math>73</math> is the same as in the sequence <math>0</math>, <math>3</math>, <math>6</math>, <math>\dotsc</math>, <math>
  2 KB (265 words) - 23:05, 26 June 2023
• 2016 AIME I Problems
  A strictly increasing sequence of positive integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>,
  8 KB (1,360 words) - 12:19, 29 January 2022
• 2015 USAJMO Problems
  ...inue with a finite sequence of moves so as to obtain in the end a constant sequence. ...ng the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
  3 KB (565 words) - 16:42, 5 August 2023
• 2015 USAJMO Problems/Problem 1
  ...inue with a finite sequence of moves so as to obtain in the end a constant sequence. Consider any arithmetic sequence. WLOG, let it be <math>s = (1, 2, 3, \dots, 2015)</math>, i.e. <math>s_i =
  8 KB (1,405 words) - 20:13, 26 July 2022
• 2015 AMC 8 Problems
  ...h row and each column in this <math>5\times5</math> array is an arithmetic sequence with five terms. The square in the center is labelled <math>X</math> as sho
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• 2015 AMC 8 Problems/Problem 18
  ...h row and each column in this <math>5\times5</math> array is an arithmetic sequence with five terms. The square in the center is labelled <math>X</math> as sho .../math> and a fifth term of <math>49</math>. The common difference of this sequence is <math>\frac{49-13}4=9</math>, so the third term is <math>13+2\cdot 9=\b
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• 2016 AMC 10B Problems
  In how many ways can <math>345</math> be written as the sum of an increasing sequence of two or more consecutive positive integers? ...ree two-digit integers <math>ab<bc<cd</math> form an increasing arithmetic sequence? One such number is <math>4692</math>, where <math>a=4</math>, <math>b=6</m
  12 KB (1,868 words) - 17:50, 30 December 2023
• 2016 AMC 10B Problems/Problem 18
  In how many ways can <math>345</math> be written as the sum of an increasing sequence of two or more consecutive positive integers? ...h>) that sums to <math>345</math>. This calls for the sum of an arithmetic sequence given that the first term is <math>k</math>, the last term is <math>g</math
  5 KB (813 words) - 16:55, 9 June 2023
• 2016 AMC 10B Problems/Problem 24
  ...ree two-digit integers <math>ab<bc<cd</math> form an increasing arithmetic sequence? One such number is <math>4692</math>, where <math>a=4</math>, <math>b=6</m To form the sequence, we need <math>(10c+d)-(10b+c)=(10b+c)-(10a+b)</math>. This can be rearrang
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• 2016 AMC 12B Problems/Problem 16
  In how many ways can <math>345</math> be written as the sum of an increasing sequence of two or more consecutive positive integers? For the first case, we can cleverly choose the convenient form of our sequence to be
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• 2016 AIME I Problems/Problem 10
  A strictly increasing sequence of positive integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, ...r sequence where <math>a_1=1</math> and <math>a_2=2</math>. Continuing the sequence,
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• 2016 AIME II Problems/Problem 9
  ....</math> are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let <math>c_n=a_n+b_n</math>. There is an integer <math>k</m ...th>1000</math>. We let <math>r</math> be the common ratio of the geometric sequence and write the arithmetic relationships in terms of <math>r</math>.
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• 2016 AIME II Problems
  ....</math> are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let <math>c_n=a_n+b_n</math>. There is an integer <math>k</m
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• Mock AIME I 2015 Problems/Problem 8
  Let <math>a,b,c</math> be consecutive terms (in that order) in an arithmetic sequence with common difference <math>d</math>. Suppose <math>\cos b</math> and <mat
  2 KB (310 words) - 19:46, 27 March 2016
• 2006 iTest Problems
  ...ath>\alpha</math> denote <math>\cos^{-1}(\tfrac 23)</math>. The recursive sequence <math>a_0,a_1,a_2,\ldots</math> satisfies <math>a_0 = 1</math> and, for all
  31 KB (4,811 words) - 00:02, 4 November 2023
• 1965 AHSME Problems/Problem 20
  ...ath>r-1</math> terms from the sum of the first <math>r</math> terms of the sequence. Plugging in <math>r</math> and <math>r-1</math> as values of <math>n</math
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• 1990 IMO Problems/Problem 4
  ...imes and their inverse. A good way to do this assuming we aren't given the sequence of all infinite primes (as this would require an explicit formula that does
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• 2016 AMC 8 Problems/Problem 19
  Note: After we combine like terms, you would have an arithmetic sequence from <math>2</math> to <math>48</math> (because <math>24 \cdot 2 = 48</math ...<math>k=200</math>. <math>2k=400</math> is the middle term of the original sequence, so the original last term is <math>400+\frac{25-1}{2}\cdot 2=424</math>. S
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• Mock AIME 4 Pre 2005/Problems
  ...3</math> and <math>a_{2004} = 7.</math> <math>b_1, b_2, \ldots</math> is a sequence of real numbers in which <math>b_n</math> is the geometric mean of the prev
  7 KB (1,094 words) - 15:39, 24 March 2019
• 2017 AMC 10A Problems/Problem 8
  ..., second, ..., ninth person. We can write this as the sum of an arithmetic sequence:
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• 1983 IMO Problems/Problem 5
  The answer is yes. We will construct a sequence of integers that satisfies the requirements.
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• 2017 AMC 8 Problems/Problem 23
  ...herefore, <math>x=5</math> because the factors have to be in an arithmetic sequence with the common difference being <math>5</math> and <math>x=5</math> is the
  2 KB (328 words) - 03:22, 16 January 2023
• 2018 AMC 10B Problems/Problem 16
  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}.<
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• 2018 AMC 10B Problems/Problem 20
  ...</math> where <math>a_n</math> represents the <math>n</math>th term of the sequence <math>A.</math> This solution will show a few different perspectives. Even To begin, we consider the sequence <math>B</math> formed when we take the difference of consecutive terms betw
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• 1960 AHSME Problems/Problem 36
  The nth term of the sequence with first term <math>a</math> is <math>a + d(n-1)</math>. That means the The 2nth term of the sequence with first term <math>a</math> is <math>a + d(2n-1)</math>. That means the
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• 1961 AHSME Problems/Problem 26
  Let the first term of the [[arithmetic sequence]] be <math>a</math> and the common difference be <math>d</math>. The <math>50^{\text{th}}</math> term of the sequence is <math>a+49d</math>, so the sum of the first <math>50</math> terms is <ma
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• 1963 AHSME Problems/Problem 16
  Let <math>d</math> be the common difference of the [[arithmetic sequence]], so <math>a = b-d</math> and <math>c = b+d</math>. ...math>1</math> or <math>c</math> by <math>2</math> results in a [[geometric sequence]],
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• 2007 iTest Problems/Ultimate Question
  ...1, 2006, 2007)</math>. A total of <math>2007</math> iterations produces a sequence with <math>2^{2007}+1</math> terms. If the integer <math>2004</math> appear
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• 2008 iTest Problems/Problem 31
  The <math>n^\text{th}</math> tern of a sequence is <math>a_n=(-1)^n(4n+3)</math>. Compute the sum <math>a_1+a_2+a_3+\cdots+ Write out each number in the sequence.
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• 2008 iTest Problems/Problem 42
  ...(positive) difference between the <math>1980^\text{th}</math> term in the sequence, and the <math>1977^\text{th}</math> term in the sequence. What number does Joshua compute?
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• 2008 iTest Problems/Problem 43
  ...si challenges her to compute the sum of all <math>2008</math> terms in the sequence. ...lexis correctly finds the sum of the <math>2008</math> terms in Dr. Lisi's sequence. What is this sum?
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• 1973 AHSME Problems/Problem 26
  Thus, there are <math>\boxed{\textbf{(E)}\ 8}</math> terms in the arithmetic sequence.
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• 1973 AHSME Problems/Problem 28
  ...<math> \log_an</math>, <math> \log_bn</math>, <math> \log_cn</math> form a sequence ...th>a</math>, <math>b</math>, and <math>c</math> are members of a geometric sequence, <math>b = ar</math> and <math>c = ar^2</math>. That means the three logar
  2 KB (277 words) - 14:04, 20 February 2020
• 1955 AHSME Problems/Problem 45
  ...first term <math>\neq 0</math> and <math>r \neq 0</math> and an arithmetic sequence with the first term <math>=0</math>. A third sequence <math>1,1,2\ldots</math> is formed by adding corresponding terms of the two
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• 2008 iTest Problems/Problem 44
  ...pose I select two distinct terms at random from the <math>2008</math> term sequence. What's the probability that their product is positive?" If <math>a</math> '''Note: Dr. Lisi’s sequence is <math>-1776, -1765, -1754 \cdots</math>'''
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• 2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1
  ...he first sequence and <math>d_2</math> the common difference in the second sequence. Thus, <math>b_4=x+4d_2</math> and <math>b_3=x+2d_2</math>. In addition, <m
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• 2019 AMC 10A Problems/Problem 15
  A sequence of numbers is defined recursively by <math>a_1 = 1</math>, <math>a_2 = \fra ...rac{1}{a_{n-2}}</math>. So <math>\{\frac{1}{a_n}\}</math> is an arithmetic sequence with step size <math>\frac{7}{3}-1=\frac{4}{3}</math>, which means <math>\f
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• 2019 AMC 10B Problems
  Define a sequence recursively by <math>x_0=5</math> and
  15 KB (2,458 words) - 23:52, 12 November 2023
• 2019 AMC 10B Problems/Problem 4
  ...he problem doesn't specify any further conditions other than an arithmetic sequence (i.e., that the numbers have to be increasing, or positive or something lik
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• What is the definition of Pure Mathematics?
  ...: inequalities, polynomials, graphing equations, arithmetic, and geometric sequence.
  35 KB (5,882 words) - 18:08, 28 June 2021
• 2020 AMC 10A Problems/Problem 8
  ...ves us <math>2+10+18...+394</math>. Clearly, this pattern is an arithmetic sequence. By using the formula we get <math>\frac{2+394}{2}\cdot 50=\boxed{\textbf{( ...h>2x</math>, and we only found the value of <math>x</math>, the sum of the sequence is <math>4950\cdot2=\boxed{\textbf{(B) } 9900}</math>.-middletonkids
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• Convergent
  ...le a arithmetic sequence that is non-infinite, or a non-infinite geometric sequence.
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• 2022 AMC 10A Problems
  ...What is the least positive integer <math>n</math> such that performing the sequence of transformations <math>T_1, T_2, T_3,...,T_n</math> returns the point <ma ...th>, <math>60</math>, and <math>91</math>. What is the fourth term of this sequence?
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• 2020 AIME II Problems/Problem 8
  Define a sequence recursively by <math>f_1(x)=|x-1|</math> and <math>f_n(x)=f_{n-1}(|x-n|)</m ...{m-1}</math>. Because the zeros of <math>f_{m-1}</math> form an arithmetic sequence with common difference <math>2,</math> so do the zeros of <math>f_m</math>.
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• 2020 FMC 10A Problems
  It is given that an arithmetic sequence <math>\{ a_n \}</math> satisfies that the initial term <math>a_0 = 0,</math ...6, 4</math> is not. The probability that Jamie will produce a fluctuating sequence by rolling the said die <math>20</math> times can be expressed as <math>\fr
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• 2020 FMC 12A Problems
  It is given that an arithmetic sequence <math>\{ a_n \}</math> satisfies that the initial term <math>a_0 = 0,</math ...6, 4</math> is not. The probability that Jamie will produce a fluctuating sequence by rolling the said die <math>20</math> times can be expressed as <math>\fr
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• 2021 AIME I Problems/Problem 5
  Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the .../math> and let the middle term be <math> x </math>. Then, we have that the sequence is
  7 KB (1,150 words) - 03:15, 1 February 2024
• 2021 AIME I Problems
  Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the Consider the sequence <math>(a_k)_{k\ge 1}</math> of positive rational numbers defined by <math>a
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• 2021 AMC 10A Problems/Problem 4
  This is an arithmetic sequence so the total distance travelled, found by summing them up is: == Video Solution (Arithmetic Sequence but in a Different Way)==
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• 2023 AIME II Problems
  The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is doub
  8 KB (1,370 words) - 21:34, 28 January 2024
• 2022 AIME I Problems/Problem 6
  ...the number of ordered pairs of integers <math>(a, b)</math> such that the sequence<cmath>3, 4, 5, a, b, 30, 40, 50</cmath>is strictly increasing and no set of ...ake, and if we take the triplet of single digit numbers, the only possible sequence must have a 6, which we already don't count. Therefore, we subtract <math>\
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• 2022 AMC 12B Problems
  How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers? ...quence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that <cmath>2^{a_7}=2^{27} \cdot a_7.</cmath> Wha
  15 KB (2,233 words) - 13:02, 10 November 2023
• 2021 AIME II Problems
  An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an a
  8 KB (1,429 words) - 14:31, 26 February 2024
• 1982 AHSME Problems/Problem 8
  ...<math>-</math> <math>{n} \choose 2,</math> because they form an arithmetic sequence, and expanding, we have by the definitions in the problem: <cmath>\frac{n(n
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• 1972 AHSME Problems/Problem 16
  ...1)}</math>, <math>\textbf{(2)}</math>, and the definition of an arithmetic sequence) <math>\textbf{(4)}</math>
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• 2021 AMC 10A Problems/Problem 22
  If we add the page numbers on each sheet, we get this sequence: The sum of the numbers in this sequence is
  10 KB (1,619 words) - 20:07, 9 November 2023
• 2022 AMC 12A Problems
  ...What is the least positive integer <math>n</math> such that performing the sequence of transformations <math>T_1, T_2, T_3, \dots, T_n</math> returns the point
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• 2021 AMC 10B Problems/Problem 15
  ...note that the exponents of <math>x^{11}-7x^7+x^3</math> are an arithmetic sequence, so they are symmetric around the middle term. So, <math>x^{11}-7x^7+x^3 =
  5 KB (806 words) - 00:13, 20 October 2023
• 2021 AMC 12B Problems/Problem 23
  "Evenly spaced" just means the bins form an arithmetic sequence. Suppose the middle bin in the sequence is <math>x</math>. There are <math>x-1</math> different possibilities for t
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• 1978 AHSME Problems/Problem 8
  WLOG, let <math>x =2</math> and <math>y = 5</math>. From the first sequence, we get From the second sequence, we get
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• 2022 USAJMO Problems
  ...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
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• 2022 AIME I Problems
  Find the number of ordered pairs of integers <math>(a,b)</math> such that the sequence <cmath>3,4,5,a,b,30,40,50</cmath> is strictly increasing and no set of four
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• 2021 JMC 10 Problems/Problem 14
  ...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 ...ath> and <math>72_b = 7b+2.</math> Because they are terms in an arithmetic sequence, the difference between <math>57_b</math> and <math>24_b</math> must be twi
  906 bytes (138 words) - 16:17, 1 April 2021
• G285 2021 MC10A
  ...th>a_{n-1} = n+a_n</math> for <math>n>1</math>. Let an infinite arithmetic sequence <math>P</math> be such that <math>P=\{k+1, k-p+1, k-2p+1 \cdots\}</math>. I Let a recursive sequence <math>a_1=1</math> and <math>a_2=13</math> be defined as: <cmath>a_n = \fra
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• Arithmetic Sequence
  #REDIRECT[[Arithmetic sequence]]
  32 bytes (3 words) - 17:05, 3 September 2021
• Arithmetic Series
  #REDIRECT[[Arithmetic sequence]]
  32 bytes (3 words) - 17:05, 3 September 2021
• 2022 AMC 10B Problems
  How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers? ...h>S_n</math> be the sum of the first <math>n</math> terms of an arithmetic sequence that has a common difference of <math>2</math>. The quotient <math>\frac{S_
  15 KB (2,224 words) - 13:10, 20 February 2024
• 2021 JMPSC Invitationals Problems/Problem 11
  Finally, the sequence of such squares is <cmath>(5\cdot 0 + 2)^2, (5\cdot 0 + 3)^2, (5\cdot 1 + 2 ...math>90^2=8100</math>. Now, note <math>92^2=8464</math> is not part of our sequence, but is the <math>37</math>th perfect square. Therefore, <math>5</math> bel
  3 KB (435 words) - 21:15, 11 July 2021
• Harmonic sequence
  ...uence is formed by taking the reciprocals of every term in an [[arithmetic sequence]]. ...stants <math>a</math> and <math>d</math>, the terms of any finite harmonic sequence can be written as <cmath>\frac{1}{a}, \textrm{ } \frac{1}{a+d}, \textrm{ }
  6 KB (1,019 words) - 16:50, 19 February 2024
• 2021 Fall AMC 12A Problems/Problem 24
  To make their lengths an arithmetic sequence, we must have <math>\theta \neq 120^\circ</math>. Because the lengths of these sides form an arithmetic sequence, we have the following system of equations:
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• Arithmetic Progression
  #REDIRECT[[Arithmetic sequence]]
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• 2022 AMC 8 Problems/Problem 6
  Let the common difference of the arithmetic sequence be <math>d</math>. Consequently, the smallest number is <math>15-d</math> a
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• Euclid 2020/Problem 4
  (b) A geometric sequence has  first term <math>10</math> and common ratio <math>1/2</math> . An arithmetic sequence has  first term <math>10</math> and common difference <math>d</math>.
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• Linear regression
  ...math>, where <math>m</math> is the number of predictors. The terms of this sequence are passed one by one into <math>f</math> as arguments; <math>f</math> is t
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• 2022 AMC 10A Problems/Problem 20
  ...th>, <math>60</math>, and <math>91</math>. What is the fourth term of this sequence? ...the arithmetic sequence be <math>a,a+d,a+2d,a+3d</math> and the geometric sequence be <math>b,br,br^2,br^3.</math>
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• 2022 AMC 10B Problems/Problem 17
  ...h>. Since <math>2^{404}+2^{202}+1\equiv0\pmod{3}</math> (Every term in the sequence is equivalent to <math>1\pmod{3}</math>), <math>2^{606}-1</math> is divisib
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• 2022 AMC 10B Problems/Problem 25
  Let <math>x_0,x_1,x_2,\dotsc</math> be a sequence of numbers, where each <math>x_k</math> is either <math>0</math> or <math>1
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• 2022 AMC 10B Problems/Problem 15
  ...h>S_n</math> be the sum of the first <math>n</math> terms of an arithmetic sequence that has a common difference of <math>2</math>. The quotient <math>\frac{S_ ...mber of terms multiplied by the median of the sequence. The median of this sequence is equal to <math>a + n - 1</math>. Thus, the value of <math>S_{n}</math> i
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• 2022 AMC 12B Problems/Problem 9
  ...quence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that <cmath>2^{a_7}=2^{27} \cdot a_7.</cmath> Wha ...ath>. To minimize <math>a_2</math>, we maxmimize <math>d</math>. Since the sequence contains only positive integers, <math>32 - 7d > 0</math> and hence <math>d
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• 2023 AMC 8 Problems/Problem 25
  ==Video Solution by SpreadTheMathLove Using Arithmetic Sequence==
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• 2023 AIME I Problems/Problem 14
  ...ally, both hands point to the number <math>12</math>. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves cloc ...umber of sequences of <math>144</math> hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the e
  11 KB (1,834 words) - 22:01, 4 January 2024
• 2023 AIME II Problems/Problem 1
  The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is doub In the arithmetic sequence, let <math>a</math> be the first term and <math>d</math> be the common diff
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• 2023 USAMO Problems/Problem 5
  ...e claim it cannot be part of a column that has cells forming an arithmetic sequence after any appropriate rearranging. After such a rearrangement, if the colum
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• 2023 AMC 10B Problems
  ...n+2} = L_{n+1}+L_n</math> for <math>n \geq 1</math>. How many terms in the sequence <math>L_1, L_2, L_3, \cdots, L_{2023}</math> are even? ...common difference <math>d>1</math>. Carl wrote down all the terms in this sequence correctly except for one term, which was off by <math>1</math>. The sum of
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• 2023 AMC 10A Problems/Problem 15
  We can see that the pattern is an arithmetic sequence with first term <math>3</math> and common difference <math>4</math>. From h We can find the sum of the first <math>32</math> terms of the arithmetic sequence by using the formula.
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• 2023 AMC 12A Problems/Problem 12
  ...we will be using difference of cube, sum of squares and sum of arithmetic sequence formulas.
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• 2023 AMC 12A Problems/Problem 20
  <math>S_n + n</math> is a geometric sequence by a ratio of <math>2</math> ...=1}^{2020}(n \cdot 2^{(2021-n)})</math>, we can solve the sum of geometric sequence to be <math>{2^{2022} - 4 - 2020 \cdot 2}</math>, which has 0 for the unit
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• 2023 AMC 10B Problems/Problem 23
  ...ommon difference <math>d > 1</math>. Carl wrote down all the terms in this sequence correctly except for one term, which was off by <math>1</math>. The sum of ...a+d(n-1)\right).</math> This can quickly be rederived by noticing that the sequence goes <math>a,a+d,a+2d,a+3d,\dots,a+(n-1)d</math>, and grouping terms.
  6 KB (1,107 words) - 09:01, 18 April 2024
• 2023 AMC 12B Problems/Problem 17
  Because three side lengths form an arithmetic sequence, the middle-valued side length is <math>\frac{x + 6}{2}</math>.
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• 2003 IMO Problems/Problem 5
  ...equality if and only if <math>x_1, x_2, ..., x_n</math> form an arithmetic sequence.
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• 2001 OIM Problems/Problem 4
  ...umber of increasing arithmetic progressions of three terms that can have a sequence <math>a_1 < a_2 < \cdots < a_n</math> of <math>n \ge 3</math> real numbers. '''Note:''' Three terms <math>a_i, a_j , a_k</math> of a sequence of real numbers form an increasing arithmetic progression if <math>a_i < a_
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• SANSKAR'S OG PROBLEMS
  ...is eventually in constant loop(i.e. after a term all the next terms in the sequence are repeating in a certain period ) Is Sanskar right? Justify your answer w For any positive integer <math>n</math>, the second or third number in the sequence will always be less than <math>n</math>.
  10 KB (1,710 words) - 08:57, 4 March 2024