Download Sequences - MazesMath

Aim: What are sequences?
Do Now:
A ball is dropped from a height of 16 feet.
Each time that it bounces, it reaches a
height that is half of its previous height.
Below is an incomplete table showing the
bounces and height. Complete the table
and then rewrite as ordered pairs.
After
Bounce
Height
(ft)
1
2
3
4
5
8
4
2
1?
?
1/2
6
7
? 1/8
?
1/4
(1, 8), (2, 4), (3, 2), (4, 1), (5, ½), (6, ¼), (7, 8)
Aim: Arithmetic Sequence
Course: Math Literacy
Sequences
Sequence – a set of numbers written is a
given order with each term of sequence
paired with a number that indicates its
position in the list.
domain 1 2 3 4 5 6 7
sequence 8 4 2 1 ½ ¼ 1/8
An infinite sequence {an} is a function
whose domain is the set of positive integers.
The function values
a1, a2, a3, a4, . . . . . an, . . .
are called terms of the sequence. If the
domain of the functions consists of the first
n positive integers only, then the sequence is
call a finite
sequence.
Aim: Arithmetic Sequence
Course: Math Literacy
Sequences
Most sequences are related by some pattern
that can be expressed by a
formula/function/expression/rule and which
will result in a value for any term, an, in the
sequence.
What is an for the following functions?
3x – 2
1/y2
x2 + 2x + 1
3n – 2
1/n2
n2 + 2n + 1
n is the variable/constant that changes in integral increments.
n=1
3n – 2
3(1) – 2
n=2
3n – 2
3(2) – 2
n=3
3n – 2
3(3) – 2
1
4
7
Aim: Arithmetic Sequence
n=4
nth
3n – 2
term
3(4) – 2
10
Course: Math Literacy
Apparent nth Term
To define a specific sequence, list the
first several terms AND the nth term.
1 1 1 1
1
, , n ,...
, , , , ,...
2 4 8 16 2
6
1 1 1 1
, ...
, , ,, , , ...
, ,
2
2 4 8 15 ( n  1)( n  n  6)
both started with the same first three
terms, but . . . .
without nth term defined, calculations
for other terms in sequence will result
in apparent nth terms.
Aim: Arithmetic Sequence
Course: Math Literacy
Finding Apparent nth Term
Find apparent nth term for
1) 1, 3, 5, 7, . . .
n:
1
2
3
4
..... n
terms: 1
3
5
7
. . . . . an
Apparent pattern: Each term is 1 less
than twice n or an = 2n - 1
2) 2/1, 3/2, 4/3, 5/4, . . .
n:
1
terms: 2/1
2
3
4
..... n
3/2
4/3
5/4
. . . . . an
Apparent pattern: Each term in
numerator is 1 more than denominator or
an = (n + 1)/n
Aim: Arithmetic Sequence
Course: Math Literacy
Aim: What are sequences?
Do Now:
Find the first four terms of the sequence
whose nth term is an = 3n - 2
Sequence – a set of numbers written is a
given order with each term of sequence
paired with a number that indicates its
position in the list.
Aim: Arithmetic Sequence
Course: Math Literacy
Finding Terms
Find the first four terms of the sequence
whose nth term is an = 3n - 2
1st term n = 1
2nd term n = 2
3rd term n = 3
4th term n = 4
a1 = 3(1) – 2 = 1
a2 = 3(2) – 2 = 4
a3 = 3(3) – 2 = 7
a4 = 3(4) – 2 = 10
Find the first four terms of the sequence
whose nth term is an = 3 + (-1)n
1st term n = 1
2nd term n = 2
3rd term n = 3
4th term n = 4
Aim: Arithmetic Sequence
a1 = 3 + (-1)1 = 2
a2 = 3 + (-1)2 = 4
a3 = 3 + (-1)3 = 2
a4 = 3 + (-1)4 = 4
Course: Math Literacy
Recursive Definition
Recursive definition of a sequence – if 1 or more
of the first terms are given – all other terms are
defined by using the previous term(s) with a
formula.
8, 4, 2, 1, ½, ¼, 1/8
a1 = 8, a2 = 4, a3 = 2, . . .
1
1
a2  a1  4   8 
2
2
1
a n  a n 1
2
Aim: Arithmetic Sequence
Course: Math Literacy
Model Problem
a) List the next three terms of the
sequence 2, 4, 8, 16, . . .
25 26 27
domain 1 2 3 4 5 6 7
2 4 8 16 a5 a6 a7
32 64 128
b) Write a general formula/expression for an
an = 2n
c) Write a recursive definition for the
sequence
a n  2a n  1
Aim: Arithmetic Sequence
Course: Math Literacy
Fibonacci Patterns
Find the next three numbers in
the sequence 1, 1, 2, 3, 5, 8, . . .
13, 21, 34, . . .
Fibonacci sequence, first published in book
titled, Liber abaci, in 1202 by Leonardo of Pisa.
Dealt with reproductive rights of rabbits.
Leonardo also introduced algebra in Europe
from the mideast. Algebra was occasionally
referred to as Ars Magna, “the Great Art”.
Aim: Arithmetic Sequence
Course: Math Literacy
Fibonacci Patterns
1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
Suppose a newly-born pair of rabbits, one male, one
female, are put in a field. Rabbits are able to mate at the
age of one month so that at the end of its second month a
female can produce another pair of rabbits. Suppose that
our rabbits never die and that the female always produces
one new pair (one male, one female) every month from the
second month on. The puzzle that Fibonacci posed was...
Aim: Arithmetic
Sequence
Course: Math Literacy
How many pairs
will there
be in one year?
Fibonacci Patterns
Recursive Rule in General Terms of a
Fibonacci-Type Sequence
sn = sn – 1 + s n – 2
Where s1 and s2 are given. The Fibonacci
sequence is that sequence for which
s1 = s2 = 1
Ex.:
a) If s1 = 5 and s2 = 2, list the first five
terms of this Fibonacci-type sequence.
5, 2, __,
__
7 __,
9 16
5+2
2+7
Aim: Arithmetic Sequence
7+9
Course: Math Literacy
Fibonacci Patterns in Nature
Spirals on scales of a pinecone:
8 right, 13 left
Aim: Arithmetic Sequence
Course: Math Literacy
Fibonacci Sequence & Golden Ratio
If we take the ratio of two successive
numbers in Fibonacci's series, (1, 1, 2, 3, 5,
8, 13, . . .) in other words, divide each
number by the number before it, we will
find the following series of numbers:
1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666...,
8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538...
approaches the
Golden Ratio - 
1 5

 1.618034
2
Aim: Arithmetic Sequence
Course: Math Literacy
Model Problem
4, 8, 12, 16, 20,
. . . .24, . . .
If the pattern is extended,
what are the next two terms?
How is this sequence
4, 8, 12, 16, 20, 24, . . .
different from the
famous Fibonacci
4 4 4 4 4
sequence?
positive integers 1
2
terms of
sequence
8 12 16
4
recursive definition?
Aim: Arithmetic Sequence
3
4
n
4n
a n  a n 1  4
Course: Math Literacy
Model Problem
Write the rule that can be used in forming a
sequence 1, 4, 9, 16, . . . , then use the rule to
find the next three terms of the sequence.
positive integers 1
2
3
4
n
terms of
sequence
4
9
16
n2
1
1, 4, 9, 16, 25, 36, 49
recursive definition? an  an1  (2n  1)
Aim: Arithmetic Sequence
Course: Math Literacy
Model Problem
Write the first five terms of the sequence
where rule for the nth term is represented by
n+2
1 2 3 4 5
n
positive integers 1 2 3 4 5
n
terms of
sequence
3
4
recursive definition?
Aim: Arithmetic Sequence
5
6 7
n+2
a n  a n 1  1
Course: Math Literacy
Model Problem
Find the first four terms of the recursive
sequence defined below.
a1 = -3 an = a(n – 1) – n
a2 = a(2 – 1) – 2
a3 = a(3 – 1) – 3
a2 = a(1) – 2
a3 = a(2) – 3
a2 = -3 – 2 = -5
a3 = -5 – 3 = -8
a4 = a(4 – 1) – 4
a4 = a(3) – 4
a4 = -5 – 4 = -12
-3, -5, -8, -12
Aim: Arithmetic Sequence
Course: Math Literacy
Model Problems
Find the next several terms, describe the
pattern.
8, 6, 7, 5, 6, 4, ___, ___, ___
Find the first three terms of the following
recursive formulas.
1
sn  1 
n
n1
sn 
n1
sn   1  n  1
n
Aim: Arithmetic Sequence
Course: Math Literacy