1. Homogenious Recurrence Relations

2. Steps for solving homogeneous linear recurrence relation:

3. Ex1) Doubling Hamster Population • Every year Stephanie’s hamster population doubles. She started with 6 hamsters. How many hamsters does she have after 7 years? After n years?

4. Ex2) Compound Interest • A person invests $1000 at 12 % interest compounded annually. Find the amount at the end of n years using a recurrence relation and initial conditions.

5. Ex3) Second-Order Linear Recurrence Relation Solve the recurrence relation .

6. EX4) Fibonacci Relation (Climbing Stairs) • An elf has a staircase of n stairs to climb. Each step it takes can cover either 1 stair or 2 stairs. • Find a recurrence relation for the number of different ways for the elf to ascent the n-staircase.

7. Non-Homogenious Recurrence Relations

8. 1.Nonhomogeneous relation:

9. Ex5)Dividing the Plane • Suppose we draw n straight lines on a piece of paper so that every pair of lines intersect, but no three lines intersect at a point. • The recurrence relation for the number of regions into which these n lines divide the plane is

10. 2. Steps for solving nonhomogeneous relation:

11. Ex6)Tower of Hanoi • There are 3 pegs mounted on a board and n disks of various sizes with holes in their centers. Initially the disks are stacked from the largest to the smallest on one peg. • The goal is to move the entire tower to another peg. The rule is • You can move only one disk at a time. • You can not place a larger disk on a smaller one. • The recurrence relation is

12. Ex7) Solve the following nonhomogeneous relation.