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• Linear recurrence
  ...ence <math>\{a_{0},a_{1},a_{2},\ldots\}</math> is said to obey a '''linear recurrence of order <math>k</math>''' if there exist constants <math>c_{0},c_{1},\ldot
  315 bytes (56 words) - 23:32, 2 June 2011

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• 2005 AIME I Problems/Problem 5
  Next, we set up a recurrence. Let <math>a_n</math> be the number of ways <math>n</math> indistinguishabl ...down however, there will only be one way to arrange the coins. We can the recurrence: <cmath>a_n=a_{n-1}+1.</cmath> Note <math>a_1=2</math>, so <math>a_8 = 9</m
  5 KB (830 words) - 01:51, 1 March 2023
• 2004 AIME I Problems/Problem 15
  After this, we can simply use the recurrence relation for <math>S_n</math>, finding
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME II Problems/Problem 15
  ...his is more easily visually demonstrated than proven.) We can repeat this recurrence, adding one every time we pair an odd to an even (but ignoring the pairing
  6 KB (899 words) - 20:58, 12 May 2022
• 1985 AIME Problems/Problem 12
  ...>th crawl (as it will be forced to move somewhere else.). Thus, we get the recurrence relation <cmath>a_n=3^{n-1}-a_{n-1}.</cmath> ...he probability that the bug is on some other vertex. We have the following recurrence relations. <cmath>A_n = \frac{1}{3}O_{n-1}</cmath> <cmath>O_n = A_{n-1} + \
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1990 AIME Problems/Problem 15
  A [[linear recurrence | recurrence]] of the form <math>T_n=AT_{n-1}+BT_{n-2}</math> will have the closed form Suppose we have such a recurrence with <math>T_1=3</math> and <math>T_2=7</math>. Then <math>T_3=ax^3+by^3=16
  4 KB (644 words) - 16:24, 28 May 2023
• 2001 AIME I Problems/Problem 14
  ...owever, things start to look messy, and it looks like we have to break our recurrence down into even smaller cases, which is something that we don't like -- we w We thus have established a recurrence relation -- since the first house either gets mail or it doesn't, and canno
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2003 AIME II Problems/Problem 3
  ...f(1) = 3</math> (A, B or C) and the answer is just <math>f(7)</math>. The recurrence relation can be found by considering the last letter of one of the valid st
  2 KB (336 words) - 17:29, 30 July 2022
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...math>. (<math>P</math> is called the ''characteristic polynomial'' of this recurrence.) ...ath>, <math>d_n=\beta^n</math> and <math>d_n=\gamma^n</math> satisfies the recurrence. Moreover, we know the following facts:
  3 KB (568 words) - 15:50, 3 April 2012
• Derangement
  From the recurrence <math>!n=n\cdot!(n-1)+(-1)^{n}</math>, we can find that <math>\lim_{n\to\in
  3 KB (473 words) - 12:57, 20 February 2024
• Mock AIME 1 Pre 2005 Problems
  A sequence <math>\{R_n\}_{n \ge 0}</math> obeys the recurrence <math>7R_n = 64 - 2R_{n-1} + 9R_{n-2}</math> for any integers <math>n \ge 2
  6 KB (1,100 words) - 22:35, 9 January 2016
• 2007 AIME I Problems/Problem 14
  ...th>n\ge 1</math>. To do this, we use the following matrix form of a linear recurrence relation ...math>n\ge 1</math>. We conclude that <math>a_n</math> satisfies the linear recurrence <math>a_{n+1}=225a_n-a_{n-1}</math>.
  13 KB (2,185 words) - 23:30, 5 December 2022
• Binet's Formula
  ...F_{n-1} + F_{n-2}.</math> This is a constant coefficient linear homogenous recurrence relation. We also know that <math>F_0 = 0</math> and <math>F_1 = 1.</math>
  3 KB (550 words) - 16:12, 24 February 2024
• 2008 AMC 12B Problems/Problem 24
  ...so far are rational, even though there is an abundance of radicals in the recurrence. This motivates us to look at our discriminants in the quadratic formula th Plugging this into our recurrence formula gives us
  9 KB (1,482 words) - 13:52, 4 April 2024
• 2007 Alabama ARML TST Problems/Problem 11
  This gives us the recurrence <math>T_n = T_{n-1} + T_{n-3}</math> for <math>n\geq 3</math>.
  2 KB (289 words) - 08:40, 26 January 2009
• Dynamical Systems
  ...rential equations]] (ordinary or partial), [[difference equations]], and [[recurrence relations]]. Almost every technical field requires the consideration of dyn
  789 bytes (107 words) - 21:52, 18 October 2017
• 1995 AHSME Problems/Problem 27
  ==Solution 3 (plain recurrence solving) == ...t guessing the form of the solution at this point we can easily solve this recurrence. Note that one can easily get rid of the "<math>+2</math>" as follows: Let
  5 KB (682 words) - 09:45, 18 February 2022
• 2006 Alabama ARML TST Problems/Problem 15
  Using this recurrence, we compute:
  3 KB (470 words) - 01:12, 28 January 2009
• 2004 AMC 12B Problems/Problem 12
  Note that the recurrence <math>a_n+a_{n+1}-a_{n+2}~=~a_{n+3}</math> can be rewritten as <math>a_n+a_ ...2-r-1 = 1</math>, so <math>(r-1)(r+1)^2 = 1</math>, so our formula for the recurrence is <math>a_n = A + (B + Cn)(-1)^n</math>.
  2 KB (364 words) - 11:41, 13 October 2021
• 2009 AMC 12A Problems/Problem 25
  The given recurrence becomes First, some intuition. The given recurrence relation: <math>a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}</math
  7 KB (990 words) - 07:23, 24 October 2022
• 2009 AIME II Problems/Problem 14
  The new recurrence then becomes <math>b_0=0</math> and <math>b_{n+1} = \frac45 b_n + \frac 35\
  4 KB (680 words) - 13:49, 23 December 2023

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