Chapter-1_Basic Structure-SEQUENCES AND SUMMATIONS.pdf
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DISCRETE MATHEMATICS I for SE Engr. Marian Mie Alimo-ot EMath 1105 Engr. Yeseil Sacramento EMath 1105 – DISCRETE MATHEMATICS I for SE CHAPTER 1: BASIC STRUCTURES...
DISCRETE MATHEMATICS I for SE Engr. Marian Mie Alimo-ot EMath 1105 Engr. Yeseil Sacramento EMath 1105 – DISCRETE MATHEMATICS I for SE CHAPTER 1: BASIC STRUCTURES SETS FUNCTIONS SEQUENCES AND SUMMATIONS MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Definition EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Definition Finite Sequence: Example: 2, 4, 6, 8, 12, 14 (there is a last number) Infinite Sequence: Example: 1, 1/2, 1/3, 1/4, 1/5, … (three dots with no number following indicate that there is no last number) EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Definition EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Definition EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Example Geometric Progression/Sequence Example: 1,2,4,8,16,32, … 2/1 = 2, 4/2 = 2, 8/4 = 2, 16/8 = 2, 32/16 = 2 The common ratio r = 2. EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Example Arithmetic Progression/Sequence Example: 11, 7, 3, -1, -5, -9 By observing: 11 – 7 = 4 7–3=4 3 – (-1) = 4 and so on… The arithmetic sequence has a common difference of 4. EMath 1105 – DISCRETE MATHEMATICS I for SE Sequences: Example EMath 1105 – DISCRETE MATHEMATICS I for SE Strings EMath 1105 – DISCRETE MATHEMATICS I for SE Recurrence Relation EMath 1105 – DISCRETE MATHEMATICS I for SE Recurrence Relation EMath 1105 – DISCRETE MATHEMATICS I for SE Recurrence Relation EMath 1105 – DISCRETE MATHEMATICS I for SE Recurrence Relation: Fibonacci Sequence EMath 1105 – DISCRETE MATHEMATICS I for SE Exercises (Sequences) Let f be the function defined by f(n) = 5n where n = (1,2,3,4,5) If f finite or infinite? What are the elements of the sequence? What are the ordered pair in f? EMath 1105 – DISCRETE MATHEMATICS I for SE Answers Let f be the function defined by f(n) = 5n where n = (1,2,3,4,5) If f finite or infinite? Finite Sequence Function What are the elements of the sequence? f=(5,10,15,20,25) What are the ordered pair in f? (1,5),(2,10),(3,15),(4,20),(5,25) EMath 1105 – DISCRETE MATHEMATICS I for SE Exercises (Sequences) Find the next term of each sequence: 1. 1, 6, 11, 16, … 21 2. 1, 8, 27, 64, … 125 3. 1, 3, 6, 10, … 15 4. 20, 17, 13, 8, … 2 5. 1, 3, 5, 7, 9, … 11 EMath 1105 – DISCRETE MATHEMATICS I for SE Exercises (Sequences) Find the 20th element of the arithmetic sequence for which the first element is 14 and the second element is 9. Let: 𝑎!" be the 20th element of the arithmetic sequence d = 9 – 14 = -5 𝑎# = 14 n = 20 𝑎!" = 𝑎# + 𝑛 − 1 𝑑 = 14 + (20-1)(-5) = -81 EMath 1105 – DISCRETE MATHEMATICS I for SE Summations EMath 1105 – DISCRETE MATHEMATICS I for SE Summations EMath 1105 – DISCRETE MATHEMATICS I for SE Summations EMath 1105 – DISCRETE MATHEMATICS I for SE Exercises (Summations) Write the following sums with sigma (Σ )notation: 𝟕 1.2 + 4 + 6 + 8 + 10 + 12 + 14 = # 𝟐𝒋 𝟔 𝒋"𝟏 2. 1 + 3 + 5 + 7 + 9 + 11 = #(𝟐𝒋 − 𝟏) 𝒋"𝟏 𝟓 ! " 3. x − x + x − x + x # $ %& = #(−𝟏)𝒋'𝟏 𝒙𝟐𝒋 𝒋"𝟏 EMath 1105 – DISCRETE MATHEMATICS I for SE Exercises (Summations) Write the expression in expanded form, then find the sum. 1. ∑+)"* −2𝑗 , 2. ∑.-"*(2𝑥 + 1) ) ∑ 3. '(%(𝑖 − 1)(2𝑖 + 1) EMath 1105 – DISCRETE MATHEMATICS I for SE REFERENCE: Kenneth H. Rosen, Discrete Mathematics and Its Applications, 8th Edition, McGrawHill, 2012
