This document discusses recursive sequence definitions, which define each term in a sequence based on the previous term using a recursion formula. It provides examples of recursive definitions and how to write explicit definitions that directly define each term without recursion. It also gives practice problems for writing recursive and explicit definitions and applying them to population growth scenarios.

1. Objectives:
1. Write and use recursive sequence
definitions.

2. What comes next?
3, 7, 11, 15, …
How do you know?

3. Recursive Definitions
Give value of tn in terms of previous
term tn – 1.
Contain two parts:
 Initial condition
 tells
where it starts
 Recursion formula
 tells
how each term is related to
previous term

4. Example
t1 = 3
tn = 2tn – 1 + 1
 t2 =
 t3 =
 t4 =

5. You Try!
Find the first 5 terms:
t1 = 10
tn = tn – 1 + n

6. Example
Give a recursive definition for:
2, 6, 18, 54, 162, …
1, 2, 6, 24, 120, 720, …

7. You Try!
Give a recursive definition for:
81, 27, 9, 3, …

8. Recursive vs. Explicit
23, 20, 17, 14, …
Recursive definition: t1 = 23
tn = tn – 1 – 3
Explicit definition: tn = 26 – 3n

9. Finding Explicit Definitions
Find the first 5 terms of
t1 = 3
tn = tn – 1 + 4
 Write an explicit definition.

10. You Try!
Write an explicit definition for
t1 = 8
tn = ½ tn – 1

11. Applications
The population of some country grows
two ways:
1. Annual growth of 1%
2. 20,000 people immigrate each year
If the population is now 5,000,000, what
will it be in 4 years?