Transcript intro to sequences and series

COMMON CORE STANDARDS
for MATHEMATICS
FUNCTIONS: INTERPRETING FUNCTIONS (F-IF)
F-IF3. Recognize that sequences are functions, sometimes defined
recursively. Whose domain is a subset of the integers.
FUNCTIONS: BUILDING FUNCTIONS (F-BF)
F-BF2. Write an arithmetic and geometric sequences both recursively
and with explicit formula, use them to model situations and translate
between the two forms.
FUNCTIONS: LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
F-LE 2. Construct linear and exponential functions, including arithmetic
and geometric sequences, given a graph, a description of a
relationship, or to input-output pairs (include reading from a table)
INTRO TO SEQUENCES AND SERIES
Guido wants to create a tile mosaic around
the Ram-Fountain. In the first week he
begins his work by placing red tiles around
the fountain as shown:
How many tiles did he add?
Figure 1
In the second week, he adds to his work by
placing purple tiles around the fountain as
shown:
How many tiles did he add?
Figure 2
In the third week, he adds to his work by
placing green tiles around the fountain as
shown:
How many tiles did he add?
Figure 3
INTRO TO SEQUENCES AND SERIES
If he continues this pattern, how many blue
tiles will he need to complete his fourth
week of work?
INTRO TO SEQUENCES AND SERIES
In the 10th week, how many tiles would you
expect him to add. How many total are
around the fountain? Explain how you
arrived at this answer.
INTRO TO SEQUENCES AND SERIES
What is a “Sequence”?
INTRO TO SEQUENCES AND SERIES
What is an “Infinite Sequence”?
An infinite sequence is a function whose domain
is the set of positive integers. The function
values
a1, a2, a3, a4, a5, a6, a7. . .
Are the terms of the sequence. If the domain of a
function consists of the first n positive integers
only, the sequence is a finite sequence
A list of numbers separated by commas:
1, 2, 4, 8...., 128………
INTRO TO SEQUENCES AND SERIES
Types of a “Sequence”?
INTRO TO SEQUENCES AND SERIES
Types of a “Sequence”?
Arithmetic: a sequence of numbers that
has a common difference (d). EX: 1, 3, 5, 7
the common difference is 2. (each term is
arrived at through addition)
INTRO TO SEQUENCES AND SERIES
Types of a “Sequence”?
Arithmetic: a sequence of numbers that
has a common difference (d). EX: 1, 3, 5, 7
the common difference is 2. (each term is
arrived at through addition)
Geometric: a sequence of numbers that
has a common ratio (r). EX: 3, 12, 48, 192
the common ratio is 4. (each term is arrived
at through multiplication)
INTRO TO SEQUENCES AND SERIES
What is a “Series”?
INTRO TO SEQUENCES AND SERIES
What is a “Series”?
A list of numbers separated by addition
signs:
1+2+4+8+....+128.
INTRO TO SEQUENCES AND SERIES
What is a “term”?
INTRO TO SEQUENCES AND SERIES
What is a “term”?
A specific number in a sequence or series.
a1= first term
a2= second term
an=nth term (or last term)
INTRO TO SEQUENCES AND SERIES
What is a “sum”?
The addition of the terms of a sequence.
S4=a1+a2+a3+a4
Sn=a1+a2+a3+...+an
***The difference between S3 and “Writing a
series of the first 3 terms” is that S3 asks
you to add the terms.
ex1.
The nth term of a sequence is given by:
an = n 2 + 2
a) Write out the first 5 terms.
ex 1 (continued)
The nth term of a sequence is given by:
an = n 2 + 2
b) What is the value of the 7th term?
ex1. (continued)
The nth term of a sequence is given by:
an = n 2 + 2
c) Find a9.
ex2.
The nth term of a sequence is given by:
an = 4(n + 2)(n – 1)
Use the table function of the graphing utility
on your calculator to write out the first 5
terms.
INTRO TO SEQUENCES AND SERIES
What is a “Recursively defined Sequence”?
A sequence in which calculating each term
is based on the value of the term before.
INTRO TO SEQUENCES AND SERIES
Recursively defined Sequence
Find the first six terms of the “famous”
sequence described below
a0  1, a1  1, ak  ak 2  ak 1
INTRO TO SEQUENCES AND SERIES
Recursively defined Sequence
Find the first six terms of the sequence
described below
a1  15,
ak 1  ak  3