CMSC 250 Discrete Structures Summation: Sequences and Mathematical Induction.

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CMSC 250 Discrete Structures Summation: Sequences and Mathematical Induction

25 June 2007Sequences & Summation2 What is Next? 2, 4, 6, 8, 10, … 1, 4, 9, 16, 25, … 2, 4, 8, 16, 32, … 0, 1, 1, 2, 3, 5, …

25 June 2007Sequences & Summation3 Sequences 2,4,6,8, … for i ≥ 1 a i = 2 i –infinite sequence with infinite distinct values For i ≥ 1 b i = (-1) i –infinite sequence with finite distinct values For 1<= i <=6 c i = i +5 –finite sequence (with finite distinct values)

25 June 2007Sequences & Summation4 Identical series?

25 June 2007Sequences & Summation5 Finding the Explicit Formula Figure the formula of this sequence Different sequences with same initial values

25 June 2007Sequences & Summation6 What is the Formula? 2, 4, 6, 8, 10, … 1, 4, 9, 16, 25, … 2, 4, 8, 16, 32, … 0, 1, 1, 2, 3, 5, …

25 June 2007Sequences & Summation7 Summation & Product Notation Sum of Items Specified Product of Items Specified

25 June 2007Sequences & Summation8 Variable ending point n as the index of the final term for n = 2 for n = 3

25 June 2007Sequences & Summation9 Telescoping Series

25 June 2007Sequences & Summation10 Factorial n! = n  (n-1)  (n-2)  …  2  1 Definition

25 June 2007Sequences & Summation11 Properties Merging and Splitting Distribution

25 June 2007Sequences & Summation12 Using the Properties

25 June 2007Sequences & Summation13 Change of Variables (1 of 2)

25 June 2007Sequences & Summation14 Change of Variables (2 of 2) Calculate new lower and upper limits –When k = 0, j = k + 1 = = 1. –When k = 6, j = k + 1 = = 7. Calculate new general term –Since j = k + 1, then k = j – 1. –Hence

25 June 2007Sequences & Summation15 Applications Indexing arrays using loops –When to start and end –… Algorithms –Convert from base 10 to base 2 –…