1. Recursion

2. Recursively defined sequences
To define a sequence recursively:
give initial conditions, i.e., the values of the first few terms explicitly;
give a recurrence relation, i.e., an equation that relates later terms in the sequence to earlier terms.
Example:
Define the following sequence recursively:
1, 4, 7, 10, 13, …
Solution: a1 = 1, an = an for n≥2

3. 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

4. Move disks from left pole to right pole
Tower of Hanoi:
Move disks from left pole to right pole
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
Pole A Pole B Pole C
Pole A Pole B Pole C

6. The Tower of Hanoi How to generalize the procedure to n disks?
How many moves are required?
Recursive procedure:
Transfer the top n-1 disks from pole A to pole B
Move the bottom disk from pole A to pole C
Transfer the top n-1 disks from pole B to pole C
Let ak 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 formula for counting the moves:
a1 = 1
an = 2an for n≥2

7. Recursive formula for Compound Interest
Suppose $10K is deposited in an account paying 3% interest compounded annually.
For each positive integer n, let
a0 = the initial amount deposited;
an = the amount on deposit at the end of year n.
Find a recursive relation for a0 , a1 , a2 ,…
assuming no additional deposits or withdrawals.
We have the following recursive formula:
a0 = 10,000
an = an *an-1 = 1.03*an-1 for n≥1

8. 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.
Examples
Recall the recursive formula for the compound interest example:
a0 = 10,000
an = 1.03*an-1 for n≥1
The explicit formula is
an = 10000*(1.03)n for n≥0
Note: this formula can be generalized to any geometric sequence.

9. Finding an explicit formula for a recursively defined sequence
Examples (cont.)
Suppose the sequence is given by the following recursive relation:
a0 = 3
an = an for n≥1
Then the explicit formula is
an = 3 + 4*n for n≥0
Note: this formula can be generalized to any arithmetic sequence.

10. Finding an explicit formula for a recursively defined sequence
Examples (cont.)
Recall the recursive formula for the Hanoi tower example:
a1 = 1
an = 2an for n≥2
How to get explicit formula?
Compute the first few terms of this sequence:
a1 = 1, a2 = 3, a3 = 7, a4 = 15, a5 = 31, a6 = 63
Based on the pattern, an = 2n – 1 for n≥1
But need a proof!
The proof by induction (blackboard)