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Google Keyword Rankings for : finding closed form series
1
https://math.stackexchange.com/questions/1068367/finding-a-closed-form-formula-for-a-sequence-that-is-defined-recursively
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https://math.stackexchange.com/questions/1068367/finding-a-closed-form-formula-for-a-sequence-that-is-defined-recursively
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2
https://discrete.openmathbooks.org/dmoi2/sec_seq-arithgeom.html
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https://discrete.openmathbooks.org/dmoi2/sec_seq-arithgeom.html
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3
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/2%3A_Sequences/2.2%3A_Arithmetic_and_Geometric_Sequences
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https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/2%3A_Sequences/2.2%3A_Arithmetic_and_Geometric_Sequences
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4
http://www3.govst.edu/wrudloff/CPSC438/CPSC438/CH05/Chapter5/Section.5.2.pdf
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http://www3.govst.edu/wrudloff/CPSC438/CPSC438/CH05/Chapter5/Section.5.2.pdf
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5
https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/sudanbachelor.pdf
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https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/sudanbachelor.pdf
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6
https://www.quora.com/What-is-the-closed-form-of-a-sequence
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https://www.quora.com/What-is-the-closed-form-of-a-sequence
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7
https://en.wikipedia.org/wiki/Closed-form_expression
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https://en.wikipedia.org/wiki/Closed-form_expression
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8
https://courses.engr.illinois.edu/cs173/fa2011/lectures/recursive-definition.pdf
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https://courses.engr.illinois.edu/cs173/fa2011/lectures/recursive-definition.pdf
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9
https://mathoverflow.net/questions/105415/what-is-the-theoretical-interest-of-finding-closed-form-solutions-of-infinite-se
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https://mathoverflow.net/questions/105415/what-is-the-theoretical-interest-of-finding-closed-form-solutions-of-infinite-se
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10
https://calcworkshop.com/combinatorics/recursive-formula/
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https://calcworkshop.com/combinatorics/recursive-formula/
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11
https://crystal.uta.edu/~gao/CSE2315_Discrete/handouts/Notes/Lect_9_Recurrence.pdf
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https://crystal.uta.edu/~gao/CSE2315_Discrete/handouts/Notes/Lect_9_Recurrence.pdf
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12
https://users.math.msu.edu/users/hensh/courses/481/summer22/quizzes/Quiz06A-Key.pdf
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https://users.math.msu.edu/users/hensh/courses/481/summer22/quizzes/Quiz06A-Key.pdf
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13
https://www.wolframalpha.com/examples/mathematics/discrete-mathematics/recurrences
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https://www.wolframalpha.com/examples/mathematics/discrete-mathematics/recurrences
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14
https://opendsa-server.cs.vt.edu/ODSA/Books/CS3/html/Summations.html
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https://opendsa-server.cs.vt.edu/ODSA/Books/CS3/html/Summations.html
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15
http://mathonline.wikidot.com/a-closed-form-of-the-fibonacci-sequence
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http://mathonline.wikidot.com/a-closed-form-of-the-fibonacci-sequence
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16
https://webpages.charlotte.edu/~hbreiter/m6105/sequences.pdf
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https://webpages.charlotte.edu/~hbreiter/m6105/sequences.pdf
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17
https://www2.math.upenn.edu/~wilf/website/GreeneWilfFibonacciSums.pdf
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https://www2.math.upenn.edu/~wilf/website/GreeneWilfFibonacciSums.pdf
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18
https://www.ms.uky.edu/~carl/ma502/html/genfun2sln1.html
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https://www.ms.uky.edu/~carl/ma502/html/genfun2sln1.html
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19
https://socratic.org/questions/question-8-is-find-a-closed-form-for-the-infinite-arithmetic-sequence
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https://socratic.org/questions/question-8-is-find-a-closed-form-for-the-infinite-arithmetic-sequence
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20
https://www.math.fsu.edu/~hoeij/issac2017.pdf
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https://www.math.fsu.edu/~hoeij/issac2017.pdf
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21
https://arxiv.org/abs/math/0405574
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https://arxiv.org/abs/math/0405574
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22
https://www.numerade.com/ask/question/starting-with-the-geometric-series-find-closed-form-when-ir-1-for-the-power-series-nx-11-x2-note-your-answer-should-be-a-function-of-that-pre-calculus-student-would-recognize-part-2-using-yo-62313/
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https://www.numerade.com/ask/question/starting-with-the-geometric-series-find-closed-form-when-ir-1-for-the-power-series-nx-11-x2-note-your-answer-should-be-a-function-of-that-pre-calculus-student-would-recognize-part-2-using-yo-62313/
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23
https://home.cs.colorado.edu/~yuvo9296/courses/csci2824/lec3-examples.html
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https://home.cs.colorado.edu/~yuvo9296/courses/csci2824/lec3-examples.html
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24
https://www.chegg.com/homework-help/questions-and-answers/part-1-finding-closed-form-expression-sum--simplify-following-products-1-r-1-r-1-2-1-r-1-r-q47207095
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https://www.chegg.com/homework-help/questions-and-answers/part-1-finding-closed-form-expression-sum--simplify-following-products-1-r-1-r-1-2-1-r-1-r-q47207095
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25
https://community.wolfram.com/groups/-/m/t/357533
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https://community.wolfram.com/groups/-/m/t/357533
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26
https://www.maplesoft.com/support/help/maple/view.aspx?path=examples/linearode&cid=885
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https://www.maplesoft.com/support/help/maple/view.aspx?path=examples/linearode&cid=885
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27
https://stackoverflow.com/questions/1041812/what-are-some-algorithms-for-finding-a-closed-form-function-given-an-integer-seq
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https://stackoverflow.com/questions/1041812/what-are-some-algorithms-for-finding-a-closed-form-function-given-an-integer-seq
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28
https://course.ccs.neu.edu/cs5002f18-seattle/lects/cs5002_lect3_fall18_notes_blank.pdf
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https://course.ccs.neu.edu/cs5002f18-seattle/lects/cs5002_lect3_fall18_notes_blank.pdf
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29
https://brilliant.org/wiki/sum-of-n-n2-or-n3/
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https://brilliant.org/wiki/sum-of-n-n2-or-n3/
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30
https://homework.study.com/explanation/how-to-find-the-closed-form-of-a-sequence.html
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https://homework.study.com/explanation/how-to-find-the-closed-form-of-a-sequence.html
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31
https://www.physicsforums.com/threads/closed-form-of-power-series.715018/
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https://www.physicsforums.com/threads/closed-form-of-power-series.715018/
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32
https://quizlet.com/explanations/questions/find-a-closed-form-for-the-geometric-series-and-determine-for-which-values-of-x-it-converges-d52187a3-dc36-49aa-a0aa-3d8a6ddb7a56
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https://quizlet.com/explanations/questions/find-a-closed-form-for-the-geometric-series-and-determine-for-which-values-of-x-it-converges-d52187a3-dc36-49aa-a0aa-3d8a6ddb7a56
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33
https://www3.cs.stonybrook.edu/~cse547/ch2hmks.pdf
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https://www3.cs.stonybrook.edu/~cse547/ch2hmks.pdf
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34
https://vadim.sdsu.edu/teaching/579/f17/HW8-solutions.pdf
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https://vadim.sdsu.edu/teaching/579/f17/HW8-solutions.pdf
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35
https://www.reddit.com/r/learnmath/comments/83tfsl/finding_closed_form_for_sequence_of_numbers/
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https://www.reddit.com/r/learnmath/comments/83tfsl/finding_closed_form_for_sequence_of_numbers/
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36
https://www.math.tamu.edu/~david.larson/pendse13.pdf
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https://www.math.tamu.edu/~david.larson/pendse13.pdf
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37
https://www.mathstat.dal.ca/~ghandeh/MATH3080/Math3080-6-sol.pdf
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https://www.mathstat.dal.ca/~ghandeh/MATH3080/Math3080-6-sol.pdf
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38
https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.933337.html
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https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.933337.html
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39
https://www.thestudentroom.co.uk/showthread.php?t=5321606
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https://www.thestudentroom.co.uk/showthread.php?t=5321606
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40
http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_discrete_f2014/ccs_discrete_f2014_lecture2.pdf
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http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_discrete_f2014/ccs_discrete_f2014_lecture2.pdf
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41
https://w3.cs.jmu.edu/spragunr/CS228_S14/activities/sequence_exercises.pdf
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https://w3.cs.jmu.edu/spragunr/CS228_S14/activities/sequence_exercises.pdf
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42
https://www.mathworks.com/matlabcentral/answers/1607600-how-to-find-closed-form-solution-involving-a-sum-over-sample-data
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https://www.mathworks.com/matlabcentral/answers/1607600-how-to-find-closed-form-solution-involving-a-sum-over-sample-data
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43
https://people.math.harvard.edu/~engelwar/MathS305/Finding%20Formulas%20for%20Linear%20Recursive%20Sequences.pdf
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https://people.math.harvard.edu/~engelwar/MathS305/Finding%20Formulas%20for%20Linear%20Recursive%20Sequences.pdf
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44
http://www.cs.yorku.ca/~gt/courses/EECS1028W20/solving-rec.pdf
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http://www.cs.yorku.ca/~gt/courses/EECS1028W20/solving-rec.pdf
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45
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8941138/
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https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8941138/
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46
https://artofproblemsolving.com/wiki/index.php/Characteristic_polynomial
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https://artofproblemsolving.com/wiki/index.php/Characteristic_polynomial
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47
https://www.ams.org/tran/2007-359-03/S0002-9947-06-03994-8/S0002-9947-06-03994-8.pdf
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https://www.ams.org/tran/2007-359-03/S0002-9947-06-03994-8/S0002-9947-06-03994-8.pdf
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48
https://scholarworks.calstate.edu/downloads/02870z516
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https://scholarworks.calstate.edu/downloads/02870z516
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49
https://www.statisticshowto.com/closed-form-solution/
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https://www.statisticshowto.com/closed-form-solution/
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50
https://math.ucr.edu/~res/math153-2019/history07a.pdf
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https://math.ucr.edu/~res/math153-2019/history07a.pdf
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51
https://www.cs.csubak.edu/~eddie/cmps2120-s19/material/ch2_sets_funcs/ch2.4.notes.html
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https://www.cs.csubak.edu/~eddie/cmps2120-s19/material/ch2_sets_funcs/ch2.4.notes.html
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52
https://proofwiki.org/wiki/Closed_Form_for_Triangular_Numbers
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https://proofwiki.org/wiki/Closed_Form_for_Triangular_Numbers
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53
http://courses.ics.hawaii.edu/ReviewICS141/morea/sequences/Sequences-QA.pdf
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http://courses.ics.hawaii.edu/ReviewICS141/morea/sequences/Sequences-QA.pdf
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54
https://saylordotorg.github.io/text_intermediate-algebra/s12-03-geometric-sequences-and-series.html
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https://saylordotorg.github.io/text_intermediate-algebra/s12-03-geometric-sequences-and-series.html
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55
https://www.math.miami.edu/~armstrong/309fa14/309hw1sol.pdf
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https://www.math.miami.edu/~armstrong/309fa14/309hw1sol.pdf
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56
https://towardsdatascience.com/why-is-the-closed-form-of-the-fibonacci-sequence-not-used-in-competitive-programming-674b805da341
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https://towardsdatascience.com/why-is-the-closed-form-of-the-fibonacci-sequence-not-used-in-competitive-programming-674b805da341
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57
https://www.math.uvic.ca/faculty/gmacgill/guide/GenFuncs.pdf
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https://www.math.uvic.ca/faculty/gmacgill/guide/GenFuncs.pdf
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58
http://www.cs.wisc.edu/wpis/papers/popl19main-p200-p-19591b3-39308-final.pdf
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http://www.cs.wisc.edu/wpis/papers/popl19main-p200-p-19591b3-39308-final.pdf
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59
https://tutorial.math.lamar.edu/classes/calcii/Series_Basics.aspx
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https://tutorial.math.lamar.edu/classes/calcii/Series_Basics.aspx
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60
https://people.math.sc.edu/girardi/m142/handouts/06FTaylorPolySeries.pdf
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https://people.math.sc.edu/girardi/m142/handouts/06FTaylorPolySeries.pdf
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61
http://www.cs.toronto.edu/~ylzhang/csc236/files/lec05-recurrence.pdf
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http://www.cs.toronto.edu/~ylzhang/csc236/files/lec05-recurrence.pdf
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62
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:constructing-arithmetic-sequences/a/converting-recursive-and-explicit-formulas-of-arithmetic-sequences
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https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:constructing-arithmetic-sequences/a/converting-recursive-and-explicit-formulas-of-arithmetic-sequences
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63
https://www.researchgate.net/figure/some-closed-form-representation-of-the-infinite-series-215-are-listed-below_tbl1_231876279
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https://www.researchgate.net/figure/some-closed-form-representation-of-the-infinite-series-215-are-listed-below_tbl1_231876279
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64
https://cs.wmich.edu/gupta/teaching/cs4310/lectureNotes_cs4310/Recurrence%20Relation%20notes.pdf
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https://cs.wmich.edu/gupta/teaching/cs4310/lectureNotes_cs4310/Recurrence%20Relation%20notes.pdf
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65
https://flylib.com/books/en/2.729.1/appendix_b_closed_form_of_a_geometric_series.html
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https://flylib.com/books/en/2.729.1/appendix_b_closed_form_of_a_geometric_series.html
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66
https://www.math.cmu.edu/~bkell/21110-2010s/formula.html
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https://www.math.cmu.edu/~bkell/21110-2010s/formula.html
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67
https://www.hindawi.com/journals/ijde/2013/929286/
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https://www.hindawi.com/journals/ijde/2013/929286/
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68
http://www.algebra.uni-linz.ac.at/people/mkauers/publications/kauers13a.pdf
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http://www.algebra.uni-linz.ac.at/people/mkauers/publications/kauers13a.pdf
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69
https://math.berkeley.edu/~shiyu/s15math53/generating_functions.pdf
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https://math.berkeley.edu/~shiyu/s15math53/generating_functions.pdf
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70
https://web.williams.edu/Mathematics/sjmiller/public_html/mathlab/public_html/jr02fall/Closed/Dan4.pdf
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https://web.williams.edu/Mathematics/sjmiller/public_html/mathlab/public_html/jr02fall/Closed/Dan4.pdf
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71
https://www.mathsisfun.com/algebra/sequences-finding-rule.html
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https://www.mathsisfun.com/algebra/sequences-finding-rule.html
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72
https://pediaa.com/what-is-the-difference-between-recursive-and-explicit/
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https://pediaa.com/what-is-the-difference-between-recursive-and-explicit/
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73
https://ximera.osu.edu/laode/linearAlgebra/closedFormSolutionsForPlanarODEs/closedFormSolutionsByTheDirectMethod
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https://ximera.osu.edu/laode/linearAlgebra/closedFormSolutionsForPlanarODEs/closedFormSolutionsByTheDirectMethod
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74
https://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class10.pdf
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https://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class10.pdf
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75
https://personal.math.ubc.ca/~sjer/math300/s6.pdf
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https://personal.math.ubc.ca/~sjer/math300/s6.pdf
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76
https://leetcode.com/problems/nth-digit/discuss/216373/four-line-python-using-closed-form-for-sum-of-digits-in-the-first-d-natural-numbers
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https://leetcode.com/problems/nth-digit/discuss/216373/four-line-python-using-closed-form-for-sum-of-digits-in-the-first-d-natural-numbers
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77
https://fabiandablander.com/r/Fibonacci.html
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https://fabiandablander.com/r/Fibonacci.html
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78
https://condor.depaul.edu/ntomuro/courses/415/notes/lecture8.html
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https://condor.depaul.edu/ntomuro/courses/415/notes/lecture8.html
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79
https://www.varsitytutors.com/hotmath/hotmath_help/topics/infinite-geometric-series
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https://www.varsitytutors.com/hotmath/hotmath_help/topics/infinite-geometric-series
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80
https://onlinetoolz.net/sequences
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https://onlinetoolz.net/sequences
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81
https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-36/issue-1/Closed-form-solutions-of-some-partial-differential-equations-via-quasi/10.1215/ijm/1255987610.pdf
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https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-36/issue-1/Closed-form-solutions-of-some-partial-differential-equations-via-quasi/10.1215/ijm/1255987610.pdf
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82
http://jwilson.coe.uga.edu/EMAT6680Fa2013/Hendricks/Essay%202/Essay2.html
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http://jwilson.coe.uga.edu/EMAT6680Fa2013/Hendricks/Essay%202/Essay2.html
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83
https://mathhelpforum.com/threads/closed-form-of-nth-partial-sum.130835/
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https://mathhelpforum.com/threads/closed-form-of-nth-partial-sum.130835/
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84
https://austinrochford.com/posts/2013-11-01-generating-functions-and-fibonacci-numbers.html
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https://austinrochford.com/posts/2013-11-01-generating-functions-and-fibonacci-numbers.html
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85
https://www.summitlearning.org/guest/focusareas/1399672?fromCourseId=245250
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https://www.summitlearning.org/guest/focusareas/1399672?fromCourseId=245250
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86
https://rgmia.org/papers/v2n6/TreesSum2.pdf
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87
https://cemc.uwaterloo.ca/events/mathcircles/2012-13/Fall/Junior78_Oct10.pdf
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88
http://www.brotherstechnology.com/docs/mi_paper1.pdf
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89
https://www.enotes.com/homework-help/find-closed-form-nth-partial-sums-determined-346679
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90
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91
https://www.waynesville.k12.mo.us/cms/lib07/MO01910216/Centricity/Domain/603/A1%20C7%20L8%20ANSWER%20KEY.pdf
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92
https://people.richland.edu/james/lecture/m116/sequences/induction.html
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93
https://ieeexplore.ieee.org/document/1278528
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94
https://www.omnicalculator.com/math/arithmetic-sequence
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95
https://www.calculator.net/number-sequence-calculator.html
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96
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