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- 31 bytes (3 words) - 20:12, 22 July 2021
- 30 bytes (3 words) - 14:49, 26 November 2021
- 30 bytes (3 words) - 14:49, 26 November 2021
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- *Division A: Series and Sequences **Regionals: Sequences (Arithmetic, Geometric, Other), explicit and recursive definitions8 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 fun4 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 constants4 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 constants4 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 seq4 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 go8 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 whic13 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 si8 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 five5 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 asterisk6 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 ther19 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 stri4 KB (772 words) - 20:09, 7 May 2024