Fall 2014 COMP 2300 Discrete Structures for Computation

Chapter 5.7 Solving Recurrence Relations by Iteration Fall 2014COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and PhysicsNorth Carolina Central University

Informally, • A set of elements written in a row and demonstrate some pattern (i.e. 1, 3, 5, 7, 9) • In the sequence denoted, • each individual elements is called a term. • in is called a subscript or index, • is the subscript of the initial term, • is the subscript of the final term. Sequences

Write the first few terms with the expectation that the general pattern will be obvious • i.e. “consider the sequence 3, 5, 7, 9 …” • Misunderstandings can occur • Give an explicit formula for it thterm, i.e. • By recursion which requires a recurrence relation. for all integers How to Express a Sequence? (Initial term) (recursive relation) for all integers

It is often helpful to know an explicit formula for the sequence • Faster computation • Proof of a theory • Such explicit formula is called a solution to the recurrence relation • The method of iteration • The most basic method for finding an explicit formula • Given a sequence defined by a recurrence relation and initial conditions, start from the initial conditions and calculate successive terms until you see a pattern How to Get an Explicit Formula?

Tips • Leave most of the arithmetic undone • Eliminate parentheses as you go from one step to the next Q: What is the explicit formula? Example

Example – cont’

Let be a fixed nonzero constant • Suppose a sequence is defined recursively as follows: • The sequence above is called a “Geometric Sequence”. • What is the explicit formula for this sequence? Explicit Formula for a Geometric Sequence

Let be a fixed nonzero constant • Suppose a sequence is defined recursively as follows: Explicit Formula for a Geometric Sequence

for all integers Two Famous Solutions

If is even • If is odd Two Famous Solutions

Explicit Formula for Tower of Hannoi Sequence

We have • My solution: • Mathematical Induction Checking the Correctness of a Formula by Mathematical Induction

We have • My solution: • Mathematical Induction Discovering That at Explicit Formula Is Incorrect

Ex 11. Ex 12. Some Examples

A bank pays interest at a rate of 6% per year compounded annually. • An denotes the amount in the account at the end of year , then for . • Assume no deposit or withdrawals during the year. • Initial amount deposited is $150,000 • Q1) How much will the account be worth at the end of 31 years? • Q2) In how many years will the account be worth $2,000,000? Some Examples

Fall 2014 COMP 2300 Discrete Structures for Computation