1. 13.2 Recursive Definitions Objective 1) Provide the recursive definition for sequences; 2) Identify the type of a sequence from a recursive definition.

2. Recursively Defined Sequences Often it is difficult to express the members of an object or numerical sequence explicitly. Example: The Fibonacci sequence: {f n } = 0,1,1,2,3,5,8,13,21,34,55,… There may, however, be some “local” connections that can give rise to a recursive definition – a formula that expresses higher terms in the sequence, in terms of lower terms.

3. Recursively Defined Sequences To define a sequence recursively, it must consists two parts: give initial condition(s), i.e., the value(s) of the first (few) term(s) explicitly (that tells where the sequence starts); give a recurrence relation, i.e., an equation that relates any term in the sequence to the preceding term(s). Example: Define the following sequence recursively: 1, 4, 7, 10, 13, … Solution: a 1 = 1, (initial condition) a n = a n–1 + 3 for n≥2(recurrence relation) Example: Recursive definition for {f n }: f 0 = 0, f 1 = 1 (initial condition) f n = f n – 1 + f n – 2 for n > 1. (recurrence relation)

4. Recursively defined sequences In 13.1 and 13.3, we have learned the explicit definitions for sequences. The same sequence can be defined explicitly as: Example: Define the following sequence explicitly: 1, 4, 7, 10, 13, … Solution: a n = 1 + (n – 1)  3 = 3n – 2 We learned from this example that it may have both explicit definition and recursive definition for the same sequence.

5. Recursion is one of the central ideas of computer science To solve a problem recursively Break it down into smaller subproblems each having the same form as the original problem; When the process is repeated many times, the last of the subproblems are small and easy to solve; The solutions of the subproblems can be woven together to form a solution to the original problem. Example: The tower of Hanoi (P. 484 #31)

6. RULES: You may only move one disk at a time. A disk may only be moved to one of the three columns. You must never place a larger disk on top of a smaller disk. INITIAL STATE GOAL STATE Tower of Hanoi: Move disks from left pole to right pole Pole A Pole B Pole C

8. The Tower of Hanoi How to generalize the procedure to n disks? How many moves are required? Recursive procedure: (1) Transfer the top n – 1 disks from pole A to pole B (2) Move the bottom disk from pole A to pole C (3) Transfer the top n – 1 disks from pole B to pole C Let a n denote the number of moves needed to transfer a tower of n disks from one pole to another using the above procedure. Then we have the following recursive relations for counting the moves: If n = 1 (or only 1 disk), then a 1 = 1 and all done.

9. The Tower of Hanoi Recursive procedure: (1) Transfer the top n – 1 disks from pole A to pole B (2) Move the bottom disk from pole A to pole C (3) Transfer the top n – 1 disks from pole B to pole C If n  2, then the recursive procedure (1) requires a n–1 moves, the recursive procedure (2) needs 1 moves, and recursive procedure (3) still takes a n–1 moves. So the recursive definition for the Tower of Hanoi is a n = a n–1 + 1 + a n–1 for n ≥ 2 or a n = 2a n–1 + 1 for n ≥ 2

10. Recursive Formula for Compound Interest Example: Suppose $10K is deposited in an account paying 3% interest compounded annually. For each positive integer n, let A 0 = the initial amount deposited; A n = the amount on deposit at the end of year n. Find a recursive relation for A 0, A 1, A 2,… assuming no additional deposits or withdrawals. We have the following recursive formula: A 0 = 10,000 A n = A n-1 + 0.03  A n-1 = 1.03  A n-1 for n≥1

11. Finding an Explicit Formula for a Recursively Defined Sequence It is often helpful to know an explicit formula for the sequence, especially if you need to compute terms with very large subscripts; to examine general properties of the sequence. Example Recall the recursive formula for the compound interest example: A 0 = 10,000 A n = 1.03  A n-1 for n≥1 The explicit formula is A n = 10000  (1.03) n for n≥0 Note: this formula can be generalized to any geometric sequence.

12. Example (cont.) Suppose the sequence is given by the following recursive relation: a 0 = 3 a n = a n-1 + 4 for n≥1 Then the explicit formula is a n = 3 + 4  n for n≥0 Note: this formula can be generalized to any arithmetic sequence. Finding an Explicit Formula for a Recursively Defined Sequence

13. Example (cont.) Recall the recursive formula for the Hanoi Tower example: a 1 = 1 a n = 2a n-1 + 1 for n≥2 How to get explicit formula? Compute the first few terms of this sequence: a 1 = 1 a 2 = 3 a 3 = 7 a 4 = 15 a 5 = 31 a 6 = 63 Based on the pattern, a n = 2 n – 1 for n≥1 The explicit definition for Hanoi Tower problem is very hard. Finding an Explicit Formula for a Recursively Defined Sequence = 2 1 – 1, = 2 2 – 1, = 2 3 – 1, = 2 4 – 1, = 2 5 – 1,= 2 6 – 1,

14. Note the difference between explicit definition and recursive definition 1.Explicit definition An explicit definition gives a n as a function of n. 2.Recursive definition Gives the initial term(s) and a recursive equation that tells how a n is related to one or more of the preceding terms.

15. One Important Note For a sequence, it may have both explicit definition and recursive definition. It also may have more than one recursive definition(equation). Examples are: 1.Suppose {a n } is an arithmetic sequence with common difference d. Then the explicit definition is a n = a 1 + (n – 1)d recursive definition 1 is a 1 = , a n = a n-1 + d recursive definition 2 is a 1 = , a 2 = , a n = 2a n-1 – a n-2 2. Suppose {b n } is a geometric sequence with common ration r. Then the explicit definition is b n = b 1 r n-1 recursive definition 1 is b 1 = , b n = rb n-1 Recursive definition 2 is

16. Example P. 482 #23. 1.Suppose that S n represents the number of dots in an n by n square array. Give a recursion equation that tells how S n+1 is related to S n. S 1 = 1, S n+1 = S n + n + n +1 S 1 = 1, S n+1 = S n + 2n + 1 for n ≥ 1 n n 1 1 1 SnSn How to get explicit formula? Compute the first few terms of this sequence: S 1 = 1 S 2 = 4 S 3 = 9 S 4 = 16 S 5 = 25 S 6 = 36 Based on the pattern, S n = n 2 for n≥1.

17. Note Some sequences may have more than one recursive definition: For example, any arithmetic sequence with common difference d can be expressed as a 1 = c a n = a n-1 + d for n≥2 Or, it may also be defined as a 1 = b, a 2 = c a n = 2a n-1 – a n-2 for n≥3.

18. Note Some sequences may have more than one recursive definition: For example, any geometric sequence with common ratio r can be expressed as a 1 = c a n = r a n-1, for n≥2 Or, it may also be defined as a 1 = b, a 2 = c a n = (a n-1 ) 2 /a n-2 for n≥3.

19. Assignment P. 481 #1 – 22, 24, 29, 30, 32