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• 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

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