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- 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
- ...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
- The increasing sequence <math>1,3,4,9,10,12,13\cdots</math> consists of all those positive ...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
- ...3, 5, 5</math>. This establishes a [[bijection]] between valid dice roll sequences and block walking paths. ...outcomes from rolling the dice. To transfer the inequality into a strictly increasing inequality, we can transform it into <math>1\leq a<b+1<c+2<d+3\leq 9</math>11 KB (1,731 words) - 02:16, 24 August 2024
- Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes <ma The number of increasing sequences of positive integers <math>a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 206 KB (923 words) - 13:17, 16 January 2007
- The number of [[increasing sequence]]s of [[positive integer]]s <math>a_1 \le a_2 \le a_3 \le \cdots \ ...han one way. For instance, another such expression for the number of such sequences is <math>{{1008 \choose 10} \choose 1}</math>. It so happens that these (w2 KB (293 words) - 15:20, 8 October 2007
- There exists a unique strictly increasing sequence of nonnegative integers <math>a_1 < a_2 < … < a_k</math> such th ...eflection across the <math>y</math>-axis. How many of the <math>125</math> sequences of three of these transformations (not necessarily distinct) will return <m13 KB (1,968 words) - 17:05, 23 November 2024
- ...possible from 0 to 9, and then find all the possible permutations of these sequences. | Common difference || Sequences possible || Number of sequences2 KB (336 words) - 12:40, 31 December 2024
- ...are different, then we count the number of strictly increasing arithmetic sequences between <math>0</math> and <math>10</math> and multiply by 2 for the decrea | Common difference || Sequences possible || Number of sequences2 KB (266 words) - 23:59, 18 October 2020
- ...sqrt{S+6ca}}, \frac{1}{\sqrt{S+6ab}} \right)</math> are similarly oriented sequences. Thus ...)=\sqrt{\frac{x^3}{x^3 + 8abc}}</math>. Then f is convex and f is strictly increasing, so by [[Jensen's inequality]] and [[AM-GM]],7 KB (1,194 words) - 05:11, 22 October 2024
- ...i}</math>, for all integers <math>1 \le i,j \le n</math>. Evidently, these sequences are similarly sorted. Then by the [[Rearrangement Inequality]], Since <math>x \mapsto \ln x</math> is a strictly increasing function, it then follows that12 KB (2,171 words) - 13:45, 11 October 2024
- ...es starting on <math>a</math>, and furthermore, the terms of each of these sequences constitute a permutation of <math>k</math> consecutive integers. ...that there are <math>2^k \cdot 2 = 2^{k+1}</math> <math>(k+1)</math>-good sequences which start on <math>a</math>. Thus the lemma holds by induction. <math>\3 KB (529 words) - 18:15, 18 July 2016
- ...a first row consisting of the odd integers <math>1,3,5,\ldots,99</math> in increasing order. Each row below the first has one fewer entry than the row above it, ...ABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14?9 KB (1,536 words) - 23:46, 25 August 2023
- : (c) they are in increasing order, left to right. ...refix sequence <math>d_1d_2d_3</math> is exactly the same as either of the sequences <math>d_4d_5d_6</math> or <math>d_5d_6d_7</math> (possibly both). Assuming15 KB (2,222 words) - 09:40, 11 August 2020
- ...>a_j = a_i + 1</math> and <math>a_k = a_j + 1</math>. Considering all such sequences <math>A</math> find the greatest value of <math>m</math>.'' ...if they belong to the same block. We can do this because it is unstrictly increasing. For example, if the sequence consists of 1000 ones, then 500 twos, then 504 KB (703 words) - 11:45, 27 November 2017
- ...ir sixth game, Michael notes to his father that the attendance seems to be increasing due to the team's great start, "There were <math>181</math> think a little deeper about math and science. One day while discussing sequences with Joshua, Dr. Lisi writes out the first <math>2008</math> terms of an ar71 KB (11,749 words) - 11:39, 20 November 2024
- Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> denote two sequences of integers defined as follows: Thus, the first few terms of the sequences are:3 KB (555 words) - 10:38, 3 June 2024
- ...th is to go from <math>(-4,-4)</math> to <math>(4,4)</math> with each step increasing either the <math>x</math>-coordinate or the <math>y</math>-coordinate by 1. Arithmetic sequences <math>\left(a_n\right)</math> and <math>\left(b_n\right)</math> have intege12 KB (1,817 words) - 14:00, 12 August 2020
- ...ath>a_j</math>'s have to be an increasing sequence. Thinking of minimizing sequences, we realize that the optimal equation is <math>a_n = \sqrt{n} - \sqrt{n-1}1 KB (210 words) - 12:30, 4 July 2013
- ...{1}{m^2}</math> <math>m-1</math> times, <math>\ldots</math>. To make other sequences, we can split <math>m-1</math> <math>\frac{1}{m^i}</math>s into <math>m(m-1 ...ps will turn <math>1</math> block into <math>2011</math> such blocks, thus increasing the total amount by <math>2010= 2 \cdot 3 \cdot 5 \cdot 67</math>. Thus <ma12 KB (1,989 words) - 15:55, 24 December 2024