Lecture 18 Section 4.1 Thu, Feb 10, 2005 Sequences Lecture 18 Section 4.1 Thu, Feb 10, 2005

Sequences Let S be a set. A finite sequence {xk} of elements of S is an ordered finite set of values x1, x2, …, xn in S. An infinite sequence {xk} of elements of S is an ordered set of values x1, x2, x3, … in S.

Sequences An example of a non-recursive sequence ak = 1/(k(k + 1)) {ak} = 1/2, 1/6, 1/12, 1/20, … An example of a recursive sequence a1 = 2, ak = 2ak – 1 – 1 for k  2. {ak} = 2, 3, 5, 9, …

Summation Notation The finite sum x1 + … + xn is represented by the expression Σk=1..n xk. The infinite sum x1 + x2 + x3 + … is represented by the expression Σk xk.

Induction and Deduction In logic, induction means to reason from the particular to the general. That is, to draw a general principle from a number of specific cases. Deduction is to reason from the general to the particular. That is, to apply a general principle to a specific case.

Using Induction Use inductive reasoning to find the value of the finite sum sn = Σk = 1..n 1/(k(k + 1)).

Example: Induction To use induction on the previous problem, we would investigate the first several cases, looking for a general pattern If n = 1, s1 = 1/2. If n = 2, s2 = 1/2 + 1/6 = 2/3. If n = 3, s3 = 2/3 + 1/12 = 3/4. If n = 4, s3 = 3/4 + 1/20 = 4/5. If n = 5, s3 = 4/5 + 1/30 = 5/6.

Example: Induction The obvious conjecture is that sn = n/(n + 1). for all integers n  1. How do we prove that this is true?

Using Induction Use inductive reasoning to find the value of the finite sum sn = Σk = 1..n 1/2k.

Example Reconsider the red dot-blue dot problem. Use the inductive method of reasoning. What if there were only one person? What if there were only two people? What if there were three people? What if there were n people, n  1?