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?