1. Leonardo Pisano Fibonacci
1.3 Sequences and Recursive Formulas
The Fibonacci Sequence
Leonardo Pisano Fibonacci

2. It was first proposed by Fibonacci
The Rabbit Problem
It was first proposed by Fibonacci
a) One pair of rabbits will reproduce after two months.
b) Each pair will give birth to a male and a female.
c) Rabbits will live indefinitely.
d) How many pairs are there after one year?

3. Month
Fibonacci Rabbits
Pairs

4. The Fibonacci Sequence
Add two consecutive terms to get the next term.
1, 1, 2, 3, 5, 8,
13,
21,
34,
55, 89, 144, …
t1 = 1,
t2 = 1,
tn = tn–1 + tn–2 , n > 2

5. The growth of certain plants13 8 5 5 4 3 3 2 2 1 1
weeks
branches

6. In a recursive sequence, a new term is generated from the previous term(s).
Example: 3, 6, 12, 24, 48, …
is a recursive sequence because each term, beginning with the second term, is the result of multiplying the previous term by two.

7. A recursive formula shows how to find each term from the previous term or terms.
Example: Determine the first four terms…
t1 = 3,
t2 = – 2,
tn = tn–1 + 3tn–2 , n > 2
t1 = 3
t3 = t3–1 + 3t3–2
t4 = t4–1 + 3t4–2
t2 = – 2
= t2 + 3t1
= t3 + 3t2
t3 = 7
= – 2 + 3(3)
= 7 + 3(–2)
= 7
t4 = 1
= 1

8. a) t1 = 3, tn = tn–1 + 2 b) t1 = 81, tn = tn–1 ÷ 3
Write the first five terms of each sequence.
a) t1 = 3, tn = tn–1 + 2 3, 5, 7, 9, 11
b) t1 = 81, tn = tn–1 ÷ 3
81,
27, 9, 3, 1
c) t1 = 2, t2 = 4, tn = 2tn–1 + 3tn–2 2, 4,
14,
40,
122

9. Write a recursive formula for each sequence:
t1 = 3, tn = tn–1 + 5
b) 2, – 6, 18, – 54, …
t1 = 2, tn = – 3tn–1
c) 2, 4, 7, 11, 16, …
t1 = 2, tn = tn–1 + n

10. A car worth $30 000 when it is new will depreciate by 20% per year.
a) What is the car worth as a percentage of the purchase price one year after it is bought?
80%
b) What is the value of the car at the end of each year for the next three years?
y3 = 15360
y1 = 24000
y2 = 19200
c) Write a recursive formula for the value of the car.
y1 = 24000,
yn = 0.8yn–1, n > 1