MCR3U1
U7L3
DISCRETE FUNCTIONS ~ SEQUENCES AND SERIES
(Recursive Formulas)
PART A ~ THE FIBONACCI SEQUENCE
In the year 1202, Italian mathematician Leonardo Pisano (nicknamed Fibonacci)
investigated a problem involving how fast rabbits could breed in ideal circumstances.
He described a situation like this:
Suppose a newly born pair of rabbits (1 male and 1 female) are put in a field. When
the rabbits are in their second month of life, they reproduce a new pair of rabbits.
Suppose that the rabbits never die and that the female always produces one new pair
(1 male and 1 female) every month and these rabbits eventually mate and so on.
If the cycle continues, how many pairs of rabbits will there be in the field?
This problem lead to the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, …
Is there a pattern in this sequence???
PART B ~ DEFINITIONS
explicit formula:
a formula that allows direct computation of any term
in a sequence
Ex.
recursive sequence:
(arithmetic sequence)
(geometric sequence)
a sequence for which the terms depend on one or more
of the previous terms
Ex.
recursive formula:
𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑
𝑡𝑛 = 𝑎(𝑟)𝑛−1
1, 1, 2, 3, 5, 8, …
(Fibonacci sequence)
a formula relating the general term of a sequence to
the previous term(s)
Ex.
MCR3U1
U7L3
PART C ~ RECURSIVE FORMULA FOR AN ARITHMETIC SEQUENCE
Ex.
Write the sequence given by the recursive formula:
𝑡1 = 5, 𝑡𝑛 = 𝑡𝑛−1 − 2, 𝑛 > 1
RECURSIVE FORMULA for an ARITHMETIC SEQUENCE
𝑡1 = 𝑎, 𝑡𝑛 = 𝑡𝑛−1 + 𝑑, 𝑛 > 1
Ex 
Determine an explicit formula and a recursive formula for the given sequence:
5, 11, 17, 23, 29, …
General Term (𝑡𝑛 )
Ex 
Recursive Formula
Determine the recursive formula for an arithmetic sequence in which 𝑡3 = 24
and 𝑡6 = 45.
MCR3U1
U7L3
PART D ~ RECURSIVE FORMULA FOR A GEOMETRIC SEQUENCE
Ex.
Write the sequence given by the recursive formula:
𝑡1 = 3, 𝑡𝑛 = −2𝑡𝑛−1 , 𝑛 > 1
RECURSIVE FORMULA for a GEOMETRIC SEQUENCE
𝑡1 = 𝑎, 𝑡𝑛 = 𝑟 𝑡𝑛−1 , 𝑛 > 1
Ex 
Determine an explicit formula and a recursive formula for the given sequence:
2, –8, 32, –128, 512, …
General Term (𝑡𝑛 )
Ex 
Recursive Formula
Determine the recursive formula for a geometric sequence where the first two
terms are 66 and 22.
HOMEWORK: p.424 #2, 6, 8ii p.430 #2, 6ii, 8 p.443 #3a