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  • *Division A: Series and Sequences **Regionals: Sequences (Arithmetic, Geometric, Other), explicit and recursive definitions
    8 KB (1,182 words) - 13:26, 3 April 2024
  • * [http://www.amazon.com/exec/obidos/ASIN/0486425665/artofproblems-20 Sequences, Combinations, Limits]
    24 KB (3,202 words) - 14:33, 13 January 2025
  • * [[Sequence | Sequences]] and [[Series]]
    2 KB (198 words) - 15:06, 7 December 2024
  • ...</math>. Generating functions allow us to represent the convolution of two sequences as the product of two power series. If <math>A</math> is the generating fun
    4 KB (659 words) - 11:54, 7 March 2022
  • A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[infinite]]. ==Monotone Sequences==
    2 KB (413 words) - 20:18, 13 November 2022
  • ...1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between consecutive terms varies. ...= a_n / a_{n-1}</math>. A similar definition holds for infinite geometric sequences. It appears most frequently in its three-term form: namely, that constants
    4 KB (649 words) - 20:09, 19 July 2024
  • ...0, 7, 14</math> and <math>4, 12, 36, 108, \ldots</math> are not arithmetic sequences, as the difference between consecutive terms varies. ...= a_n - a_{n-1}</math>. A similar definition holds for infinite arithmetic sequences. It appears most frequently in its three-term form: namely, that constants
    4 KB (736 words) - 01:00, 7 March 2024
  • ...we still write <math>\{a_i\}\succ\{b_i\} </math> if this is true after the sequences have been sorted in nonincreasing order.
    2 KB (288 words) - 21:48, 5 July 2023
  • Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = ...isely thus: Let <math>\{ \{a_{ij}\}_{i=1}^n \} _{j=1}^m</math> be several sequences of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a seq
    4 KB (774 words) - 11:12, 29 October 2016
  • The sums alternate between separate arithmetic sequences. One of them is of particular interest to us: specifically, the one that go
    8 KB (1,334 words) - 16:37, 15 December 2024
  • Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in whic
    13 KB (1,957 words) - 11:53, 24 January 2024
  • ...ath>B</math>, and the difference between the greatest terms of the two two sequences is <math>99</math> (forget about absolute value, it's insignificant here si
    8 KB (1,431 words) - 16:50, 29 December 2024
  • Let's write the sequences of <math>n</math>'s in a compact form , <math>T_p = 24 + 22 + 9(p-1)</math> Let's also write the sequences of <math>n</math>'s in a compact form , <math>T_q = 21 + 43 + 9(q-1)</math>
    11 KB (1,857 words) - 11:57, 18 July 2024
  • ...re two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five
    5 KB (847 words) - 23:35, 19 December 2024
  • ...are shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk
    6 KB (902 words) - 07:57, 19 June 2021
  • ...terms are formed by multiplying the corresponding terms of two arithmetic sequences.
    7 KB (1,127 words) - 08:02, 11 July 2023
  • ...s with 1 is <math>\frac{1}{10}</math>. This means we can find all possible sequences with one digit repeated twice, and then divide by <math>10</math>. ...with <math>a</math> repeated twice, we can make a table with all possible sequences:
    5 KB (855 words) - 19:26, 14 January 2023
  • ...<math>\bigl( 1/(y+z), 1/(z+x), 1/(x+y) \bigr)</math> are similarly sorted sequences, it follows from the [[Rearrangement Inequality]] that ...nd <math>\bigl(x/(y+z), y/(z+x), z/(x+y)\bigr)</math> are similarly sorted sequences. Then by [[Chebyshev's Inequality]],
    7 KB (1,164 words) - 00:01, 26 July 2024
  • == Solution 8 (Sequences) == How many distinct sequences of <math>8</math> integers <math>a_1, a_2, a_3, \ldots, a_8</math> are ther
    19 KB (3,128 words) - 20:38, 23 July 2024
  • ...>, four <tt>TH</tt>, and five <tt>TT</tt> subsequences. How many different sequences of 15 coin tosses will contain exactly two <tt>HH</tt>, three <tt>HT</tt>, Let's consider each of the sequences of two coin tosses as an [[operation]] instead; this operation takes a stri
    4 KB (772 words) - 20:09, 7 May 2024

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