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2. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: If you have any questions or problems please email: educators@boundless.com http://boundless.com/teaching-platform

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4. Introduction to Sequences Finding the General Term Sums and Series Notation: Sigma Recursive Definitions Sequences and Series Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Introduction to Sequences Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series- 53/introduction-to-sequences-224- 5904?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Sequence View on Boundless.com Sequences, Series, and Combinatorics > Sequences and Series

6. An arithmetic sequence is arrived at by adding a constant to the previous term of a sequence to arrive at the next term. It can be described by the formula [Equation 1]. Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series- 53/introduction-to-sequences-224- 5904?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 1 View on Boundless.com

7. A geometric sequence is one in which the previous member of a sequence is multiplied by a constant to arrive at the next term. It can be described by the formula [Equation 2]. Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series- 53/introduction-to-sequences-224- 5904?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 2 View on Boundless.com

8. Plotting points on a Cartesian coordinate plane and using graphing software to solve is the easiest and most effective way to find a general term. Given terms x1, x2... xn, convert these to points (1, x1), (2, x2)... (n, xn) and plot. Then find the best-fit curve. By hand, one can take the differences between each term, then the differences between the differences in terms, etc, until each difference becomes constant. Once a constant difference is achieved, one can work backwards to find the relationship among terms in each set of the differences. Eventually, one can work backwards to obtain a sum for the initial series. Finding the General Term Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/finding-the- general-term-225- 11211?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Sequences, Series, and Combinatorics > Sequences and Series

9. A series is merely the sum of the terms of a series. The notation for this operation is to use the capital greek letter sigma, following the general formula: [Equation 3]. Sums and Series Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/sums-and- series-226- 5544?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 3 View on Boundless.com Each person infects two more people with the flu virus View on Boundless.com

10. The sum of an arithmetic series can be calculated using the equation [Equation 4]. Sums and Series Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/sums-and- series-226- 5544?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 4 View on Boundless.com Each person infects two more people with the flu virus View on Boundless.com

11. The sum of a geometric series can be calculated using the equation: [Equation 5]. Sums and Series Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/sums-and- series-226- 5544?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 5 View on Boundless.com Each person infects two more people with the flu virus View on Boundless.com

12. A summation is performed on a series, or list of numbers. Each term is added to the next, resulting in a sum of all terms. Sigma notation is used to represent the summation of a series. In this form, the capital Greek letter sigma (Σ) is used. The range of terms in the summation is represented in numbers below and above the Σ; the lowest term is written below and the greatest term is written above. Less common forms of sigma notation often leave out specific components of the notation, such as the upper and lower boundaries for the summation to occur—if these boundaries are implicit in the context of the mathematical problem. Notation: Sigma Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/notation- sigma-227- 5838?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Summations Involving Exponential Terms View on Boundless.com Sequences, Series, and Combinatorics > Sequences and Series

13. In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself. Most recursive definitions have three foundations: a base case (basis), an inductive clause, and an extremal clause. Often only the inductive clause is discussed, which is the general case, and then as an example one will be asked to demonstrate the inductive clause given a base case or basis. Recursive Definitions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/recursive- definitions-228- 8430?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess The flue virus is a geometric sequence View on Boundless.com Sequences, Series, and Combinatorics > Sequences and Series

14. The recursive definition for an arithmetic sequence is: [Equation 6]. The recursive definition for a geometric sequence is: [Equation 7]. Again, plugging in an initial value (n_1=1 for example} will allow the nth term of sequence to be calculated, if r or d is known. Sequences, Series, and Combinatorics > Sequences and Series Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/sequences-series-and-combinatorics-8/sequences-and-series-53/recursive- definitions-228- 8430?campaign_content=book_196_section_53&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 6 View on Boundless.com Equation 7 View on Boundless.com

15. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

16. Key terms arithmetic Of a progression, mean, etc, computed using addition rather than multiplication. finite Limited, constrained by bounds, impermanent. general term A mathematical expression containing variables and constants that, when substituting integer values for each variable, produces a valid term in a sequence or series. geometric increasing or decreasing in a geometric progression, i.e. multiplication by a constant. infinite regress A regress into an infinite sequence of propositions in an attempt to found the truth of the proposition Pi on the truth of the proposition Pi+1. sequence A set of things next to each other in a set order; a series series The sum of the terms of a sequence. set A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it. sigma The symbol Σ, used to indicate summation of a set or series. summation A series of items to be summed or added. vector A directed quantity with both magnitude and direction. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics

17. Interactive Graph: Series Points Fit a Trendline From the data points in the series, a trendline with the equation is generated. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-bff67510-9251-440f-9dc8-4b0c92adff49 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-bff67510-9251-440f-9dc8-4b0c92adff49View on Boundless.com Sequences, Series, and Combinatorics

18. Sequence Part of an infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is, however, bounded within the two dashed lines. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Cauchy sequence illustration2." CC BY-SA http://en.wikipedia.org/wiki/File:Cauchy_sequence_illustration2.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Cauchy_sequence_illustration2.svgView on Boundless.com Sequences, Series, and Combinatorics

19. The flue virus is a geometric sequence Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Sequences and series: Arithmetic & Geometric Sequences, Recursive Formulae (Grade 12)." CC BY 3.0 http://cnx.org/content/m39302/latest/#uid16 View on Boundless.comCC BY 3.0http://cnx.org/content/m39302/latest/#uid16 View on Boundless.com Sequences, Series, and Combinatorics

20. Each person infects two more people with the flu virus So the total number of people infected follows a geometric series. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Sequences and series: Arithmetic & Geometric Sequences, Recursive Formulae (Grade 12)." CC BY 3.0 http://cnx.org/content/m39302/latest/ View on Boundless.comCC BY 3.0http://cnx.org/content/m39302/latest/View on Boundless.com Sequences, Series, and Combinatorics

21. Summations Involving Exponential Terms Formulas relating to summations that you may later encounter in math classes, presented here as a reference. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "∑." GNU FDL http://en.wikipedia.org/wiki/%25E2%2588%2591 View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/%25E2%2588%2591View on Boundless.com Sequences, Series, and Combinatorics

22. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Which of the following sequences represents an arithmetic sequence? A) 2,4,8,16... B) 2,4,6,8,... C) 2,6,18,54... D) 6,3,3/2,3/4,...

23. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Which of the following sequences represents an arithmetic sequence? A) 2,4,8,16... B) 2,4,6,8,... C) 2,6,18,54... D) 6,3,3/2,3/4,...

24. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Which of the following represents a geometric sequence? A) 1,2,4,8,... B) 1,2,3,4,... C) 1,3,5,7,... D) 1,3/2,2,5/2,...

25. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Which of the following represents a geometric sequence? A) 1,2,4,8,... B) 1,2,3,4,... C) 1,3,5,7,... D) 1,3/2,2,5/2,...

26. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Which of the following represents an example of a finite sequence? A) The set of prime numbers. B) None of these answers. C) The set of odd numbers. D) A geometric sequence starting at 3, with a common ratio of 2.

27. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Which of the following represents an example of a finite sequence? A) The set of prime numbers. B) None of these answers. C) The set of odd numbers. D) A geometric sequence starting at 3, with a common ratio of 2.

28. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Find the formula for the general term n of the following sequence: 7, 14, 25, 40, 59,... A) f(n)=2x B) f(n)=2x2+x+4 C) f(n)=2x3+x+4 D) f(n)=2x2+x-4

29. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Find the formula for the general term n of the following sequence: 7, 14, 25, 40, 59,... A) f(n)=2x B) f(n)=2x2+x+4 C) f(n)=2x3+x+4 D) f(n)=2x2+x-4

30. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Find the formula for the general term n of the following sequence: 3, -5, -19, -39, -65,... A) f(n)=-3n2+n+5 B) f(n)=3n2+n+5 C) f(n)=-3n2-n+5 D) f(n)=-3n2+n-5

31. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Find the formula for the general term n of the following sequence: 3, -5, -19, -39, -65,... A) f(n)=-3n2+n+5 B) f(n)=3n2+n+5 C) f(n)=-3n2-n+5 D) f(n)=-3n2+n-5

32. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics If the first term of a geometric series is 7, and the common ratio is 3, what is the sum of the first 7 terms of the series? A) 1093 B) 2187 C) 7651 D) 5432

33. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics If the first term of a geometric series is 7, and the common ratio is 3, what is the sum of the first 7 terms of the series? A) 1093 B) 2187 C) 7651 D) 5432

34. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics If an arithmetic series has 10 terms, the first term is 3, and the last term is 57, what is the sum of the series? A) 600 B) 300 C) 450 D) 500

35. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics If an arithmetic series has 10 terms, the first term is 3, and the last term is 57, what is the sum of the series? A) 600 B) 300 C) 450 D) 500

36. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Find the fourth term in the sequence that starts with a1=-4 and is defined by the following recursive formula: an=an-1+5. A) 6 B) 11 C) 16 D) 1

37. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Find the fourth term in the sequence that starts with a1=-4 and is defined by the following recursive formula: an=an-1+5. A) 6 B) 11 C) 16 D) 1

38. Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics Find the recursive formula that defines that following sequence: 2, 5, 26, 677... A) an=(an-1)2+1 B) an=(an-1)2-1 C) an=(an-1)3+1 D) an=(an-1)3-1

39. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Sequences, Series, and Combinatorics Find the recursive formula that defines that following sequence: 2, 5, 26, 677... A) an=(an-1)2+1 B) an=(an-1)2-1 C) an=(an-1)3+1 D) an=(an-1)3-1

40. Attribution Wikipedia. "Summation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Summation#Capital-sigma_notationCC BY-SA 3.0http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation Connexions. "Series." CC BY 3.0 http://cnx.org/content/m19074/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m19074/latest/?collection=col10624 Wikipedia. "Summation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Summation#Capital-sigma_notationCC BY-SA 3.0http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation Wiktionary. "arithmetic." CC BY-SA 3.0 http://en.wiktionary.org/wiki/arithmeticCC BY-SA 3.0http://en.wiktionary.org/wiki/arithmetic Wiktionary. "geometric." CC BY-SA 3.0 http://en.wiktionary.org/wiki/geometricCC BY-SA 3.0http://en.wiktionary.org/wiki/geometric Connexions. "Series." CC BY 3.0 http://cnx.org/content/m19074/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m19074/latest/?collection=col10624 Wikipedia. "Summation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Summation#Capital-sigma_notationCC BY-SA 3.0http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation Wiktionary. "vector." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vectorCC BY-SA 3.0http://en.wiktionary.org/wiki/vector Wiktionary. "sigma." CC BY-SA 3.0 http://en.wiktionary.org/wiki/sigmaCC BY-SA 3.0http://en.wiktionary.org/wiki/sigma Wiktionary. "summation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/summationCC BY-SA 3.0http://en.wiktionary.org/wiki/summation Connexions. "Sequences and series: Arithmetic & Geometric Sequences, Recursive Formulae (Grade 12)." CC BY 3.0 http://cnx.org/content/m39302/latest/CC BY 3.0 http://cnx.org/content/m39302/latest/ Wikipedia. "Sequence." CC BY-SA 3.0 http://en.wikipedia.org/wiki/SequenceCC BY-SA 3.0http://en.wikipedia.org/wiki/Sequence Connexions. "Sequences." CC BY 3.0 http://cnx.org/content/m19076/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m19076/latest/?collection=col10624 Wiktionary. "finite." CC BY-SA 3.0 http://en.wiktionary.org/wiki/finiteCC BY-SA 3.0http://en.wiktionary.org/wiki/finite Wiktionary. "set." CC BY-SA 3.0 http://en.wiktionary.org/wiki/setCC BY-SA 3.0http://en.wiktionary.org/wiki/set Wiktionary. "sequence." CC BY-SA 3.0 http://en.wiktionary.org/wiki/sequenceCC BY-SA 3.0http://en.wiktionary.org/wiki/sequence Wikipedia. "Recursive definition." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Recursive_definitionCC BY-SA 3.0http://en.wikipedia.org/wiki/Recursive_definition Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics

41. Wiktionary. "infinite regress." CC BY-SA 3.0 http://en.wiktionary.org/wiki/infinite+regressCC BY-SA 3.0http://en.wiktionary.org/wiki/infinite+regress Connexions. "Sequences and series: Arithmetic & Geometric Sequences, Recursive Formulae (Grade 12)." CC BY 3.0 http://cnx.org/content/m39302/latest/CC BY 3.0 http://cnx.org/content/m39302/latest/ Wikipedia. "Series (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Series_(mathematics)CC BY-SA 3.0http://en.wikipedia.org/wiki/Series_(mathematics) Wikipedia. "Sequence." CC BY-SA 3.0 http://en.wikipedia.org/wiki/SequenceCC BY-SA 3.0http://en.wikipedia.org/wiki/Sequence Wiktionary. "series." CC BY-SA 3.0 http://en.wiktionary.org/wiki/seriesCC BY-SA 3.0http://en.wiktionary.org/wiki/series Wiktionary. "sequence." CC BY-SA 3.0 http://en.wiktionary.org/wiki/sequenceCC BY-SA 3.0http://en.wiktionary.org/wiki/sequence Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/general-termCC BY-SA 3.0http://www.boundless.com//algebra/definition/general-term Free to share, print, make copies and changes. Get yours at www.boundless.com Sequences, Series, and Combinatorics