MAT 251 Discrete Mathematics

1. MAT 251 � Discrete Mathematics Recurrence Relations

2. Section 7.1 Recurrence Relations Def: A sequence is a function from a subset of the set of integers (usually the set {0, 1, 2, ..} or the set {1, 2, 3 �} to a set S. To distinguish these from functions on R, we use the notation an to denote the image of the integer n.

3. Section 7.1 Recurrence Relations Examples: 1) {1, 2, 3, �} = {n}8n=1 2) { 0.1, 0.01, 0.001, �} = {.1n } 8n=1 3) {0, 1, 1, 2, 3, 5, 8, �} = ?

4. Section 7.1 Recurrence Relations The example 3 is a very special sequence called the Fibonocci sequence. It is a bit a challenge to find a closed form of this function, however, we can make a neat observation. {0, 1, 1, 2, 3, 5, 8, �}

5. Section 7.1 Recurrence Relations This is an example of a definition that we call recursive or inductive definition, that is a function that is defined by a) specifying the value of the function at n = 0, (or for n = 0 and n = 1 � n = k) (initial condition of the function) b) then giving a rule for finding the value of the function at an integer from its values at smaller integers. (This formula is called the recurrence relation itself.)

6. Section 7.1 Recurrence Relations For the Fibonocci Sequence it works like this: f0 = 0, f1 = 1, and fn = fn-1 + fn-2 Use the above definitions to find the f8.

7. Section 7.1 Recurrence Relations Another nice example is the factorial function. 0! = 1 1! = 1 n! = n*(n-1)* �* 2*1

8. Section 7.1 Recurrence Relations Example: Find the first five terms of the sequence defined by: a0 = 3, a1 = -1 and an = 6an-1 - 2an-2

9. Section 7.1 Recurrence Relations Often we are interested in looking at the recurrence relations alone and seeing what sequences if any will satisfy them. Such as how many different sequences will satisfy the RR? an = 3an-1 + an-2 � 5an-4 Another question that people often are interested in looking at is the reverse. Given a sequence what recurrence relations will they satisfy? Example: Can we find a RR for the sequence an = 2n + 1?

10. Section 7.1 Recurrence Relations A sequence is said to be a solution of a recurrence relation provided that its terms satisfy the recurrence relation. How do we determine that? Example: Is an = 4n a solution of the recurrence relation an = -3an-1 + 4an-2?

11. Section 7.1 Recurrence Relations Note that when we are given Initial conditions, that is we are given the values of the terms that are needed for the recurrence relation; and a recurrence relation. Then we have a uniquely defined sequence. (Can think of it as a unique solution to the RR ?)

12. Section 7.1 Recurrence Relations How can we used this to model real life situations? Example: Suppose that the number of bacteria in a colony quadruples every hour. a) Set up a recurrence relation for the number of bacteria after n hours. b) Find an explicit formula for the number of bacteria in a colony after n hours. c) If there were 500 bacteria initially, how many bacteria will be in the colony after 10 hours?

13. Section 7.1 Recurrence Relations Example: Suppose that Don graduates in May of 2008 and his grandparents give him $25,000 as a graduation present. Don is very happy but his grandparents insist that he deposit this money in a savings account which yields 8% annually for at least 20 years. a) Set up a recurrence relation for the amount of money in the account after n years. b) Find an explicit formula for the amount of money in the account after n years. c) How much money will Don have if he is able to withdraw after 20? What if he waits another 5 years?

14. Section 7.1 Recurrence Relations The notes have been created using K. H. Rosen�s Discrete Mathematics and Applications, 6th Edition