9.1 Mathematical Patterns
Sometimes steps in a process form a
pattern. You can describe some patterns
with a sequence, or ordered list of numbers.
Each number in a sequence is a term.
Example: Suppose you drop a ball from a
height of 10 feet, after each time it hits the
ground it rebounds 85% of its previous
height. How high will it rebound on its
fourth bounce?
10ft bounce (.85) * 10
8.5 ft bounce (.85) * 8.5
7.225 ft bounce (.85) * 7.225
6.141 ft bounce (.85) * 6.141
5.220 feet on 4th bounce
We can use a variable, such as a, with a
positive integer subscript to represent the
terms in a sequence.
1st term
a1
… n-1 term
an-1
2nd term
a2
nth term
an
3rd term
a3
n+1 term
an+1
A recursive formula defines the terms in a
sequence by relating them to the term
before. You always give the initial value.
The first example was recursive because the
next height was related to the previous
height.
The formula for example one is:
an = 0.85an-1
where a1 = 10
A formula that expresses the nth term in
terms of n is an explicit formula.
Example: Write an explicit formula for each
sequence. Find a12.
4,5,6,7,8,….
The formula is an = n+3
a12 = 15
Write a recursive formula?
an = an-1 +1
a1 = 4
CW pg 569 7 – 14, 15 – 35 odds