1 / 11

Mathematical Induction II

Mathematical Induction II. Lecture 22 Section 4.3 Mon, Feb 26, 2007. Example. Find a formula for 1 + 3 + 5 + … + (2 n – 1). and use mathematical induction to prove that it is correct. Exercise. Find a formula for 1 2 + 3 2 + 5 2 + … + (2 n – 1) 2 .

farren
Download Presentation

Mathematical Induction II

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathematical Induction II Lecture 22 Section 4.3 Mon, Feb 26, 2007

  2. Example • Find a formula for 1 + 3 + 5 + … + (2n – 1). and use mathematical induction to prove that it is correct.

  3. Exercise • Find a formula for 12 + 32 + 52 + … + (2n – 1)2. and use mathematical induction to prove that it is correct.

  4. Example: Binary Search Trees • In a complete binary search tree of depth n - 1, (n rows) what is the average number of comparisons required to locate an element? (Assuming all positions are equally likely.) • Analysis • Elements in row 1 require 1 comparison. • Elements in row 2 require 2 comparisons. • Elements in row 3 require 3 comparisons. • In general, elements in row k require k comparisons.

  5. Example: Binary Search Trees • Further analysis • Row 1 contains 1 element. • Row 2 contains 2 elements. • Row 3 contains 4 elements. • In general, row k contains 2k - 1 elements. • Fact • The average number of comparisons is the total number of comparisons required for all elements divided by the number of elements.

  6. Example: Binary Search Trees • The total number of elements is 1 + 2 + 4 + 8 + … + 2n – 1. • The total number of comparisons is 11 + 22 + 34 + 48 + … + n2n – 1. • Therefore, the average is

  7. Example: Binary Search Trees • Prove by mathematical induction that 1 + 2 + 4 + 8 + … + 2n – 1 = 2n – 1. • Prove by mathematical induction that 11 + 22 + 34 + 48 + … + n2n – 1 = (n – 1)2n + 1. • Then, the average is • This is approximately equal to n – 1 (for large n).

  8. Example: Recursive Sequences • Define a sequence a1 = 2, an = 2an – 1 – 1 for n  2. • Find a non-recursive formula for an and prove that it is correct. • Analysis • {an} = {2, 3, 5, 9, 17, 33, …}. • {an – 1} = {1, 2, 4, 8, 16, 32, …}. • Conjecture that an = 2n – 1 + 1.

  9. Example: Recursive Sequences • Proof: • Basic Step • P(1) is true since a1 = 2 = 20 + 1. • Inductive Step • Suppose that ak = 2k – 1 + 1 for some k 1.

  10. Example: Recursive Sequences • Then ak + 1 = 2ak – 1 (by def.) = 2(2k – 1 + 1) – 1 (by ind. hyp.) = 2k + 2 – 1 = 2k + 1. • Therefore, P(k + 1) is true. • Therefore, an = 2n + 1 for all n 1.

  11. Example • Define {an} as • a1 = 2, • an = 3an – 1 + 1, for all n 2. • Find a non-recursive formula for an and prove that it is correct using mathematical induction.

More Related