8.1 sequences comp.notebook
January 10, 2017
8­1 Sequences
Defining a Sequence
A sequence {an} is a list of numbers written in explicit order.
e.g. If the domain is finite, then the sequence is a
Ex.1 Explicit Sequence
Find the first six terms and the 100th term of the sequence {an}, where This example is defined because it is defined in terms of n. Dec 27­3:54 PM
8.1 sequences comp.notebook
January 10, 2017
Recursive Sequences depend on what has gone on before.
Ex. 2
Find the first three terms and the sixth term for the recursive sequence defined by the following conditions:
b1 =5
bn=2bn­1 ­ 3 for ∀n≥2
TRY
Repeat example 2 with the following conditions:
b1 = ­ 4
b2 = 7
bn=3bn­2 + 2bn­1 Dec 27­4:24 PM
for ∀n≥3
8.1 sequences comp.notebook
January 10, 2017
Arithmetic Sequences
Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d.
d is the Recursive definition: Explicit definition:
Dec 27­5:43 PM
8.1 sequences comp.notebook
January 10, 2017
Ex. 3 Defining Arithmetic Sequences
For the arithmetic sequence {­7,­3,1,5,9,...} find:
(a) the common difference
(b) a recursive rule for the nth term
(c) an explicit rule for the nth term
(d) the 42nd term
Ex.4 Repeat the above for the arithmetic sequence {ln2, ln6, ln18, ln54,...}
Jan 4­9:22 AM
8.1 sequences comp.notebook
January 10, 2017
Ex. 5 Constructing a Sequence
The third and sixth terms of an arithmetic sequence are 5 and 14, respectively. Find the common difference, first term, and an explicit rule for the nth term.
Jan 4­9:34 AM
8.1 sequences comp.notebook
January 10, 2017
Geometric Sequences
Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r.
r is the Recursive definition: Explicit definition:
Dec 27­5:43 PM
8.1 sequences comp.notebook
January 10, 2017
Ex. 6 Defining Geometric Sequences
For the geometric sequence {4,­12,36,­108,...} find:
(a) the common ratio
(b) a recursive rule for the nth term
(c) an explicit rule for the nth term
(d) the seventh term
Ex. 7
Repeat example 6 for the geometric sequence {5­3 ,5­5 ,5­7 ,5­9 ,...}
Jan 4­9:22 AM
8.1 sequences comp.notebook
January 10, 2017
Ex. 8 Constructing a Sequence
The third and sixth terms of a geometric sequence are ­20 and 160, respectively. Find the common ratio, first term, and an explicit rule for the nth term.
Jan 4­9:34 AM
8.1 sequences comp.notebook
January 10, 2017
Limit of a Sequence
Some sequences do not have a limit e.g.
Some sequences tend towards a limit as n­>
∞ e.g. Notation: L is the limit of the sequence. If L exists we say the sequence CONVERGES to L. Sequences that do not have limits DIVERGE.
Properties of Limits
If L and M are real numbers and and , then
1) Sum Rule: 2) Difference Rule: 3) Product Rule: 4) Quotient Rule: 5) Constant Multiple Rule: Jan 4­10:20 AM
8.1 sequences comp.notebook
January 10, 2017
Ex. 9 Finding the Limit of a Sequence
Determine whether the sequence converges or diverges. If it converges, find its limit.
(a) Ex. 10
(b) Determining Convergence or Divergence
Determine whether the sequence with the given nth term converges or diverges. If it converges, find its limit.
(a) n=1,2,3,... (b) b1=4, bn=bn­1 +2 ∀n≥2
Jan 4­10:30 AM
8.1 sequences comp.notebook
January 10, 2017
Sandwich Theorem for Sequences
If and there is an integer N for which an≤bn≤cn for ∀n>N, then .
Ex. 11
Show that the sequence converges, and find its limit.
Ex. 12
Determine if the sequence converges, if it converges find its limit.
Factorials
In your groups:
Evaluate 4!
Expand n! 3 different ways.
What is ?
What is ?
What is ?
Which one grows faster?
What is ?
Which one gets smaller more quickly?
Jan 4­10:40 AM
8.1 sequences comp.notebook
January 10, 2017
Ex.13 Graphing using Parametric Mode
Sometimes it helps to represent a geometric sequence graphically. It is a good idea to use parametric mode when graphing sequences.
Draw a graph of the sequence {an} with a n=
Solution
Let X1=T and Y1=(­1)T(T­1)2/T Graph in dot mode.
Tmin=1 and Tmax=20 with TStep=1
Xmin=0 and Xmax=20 with XScl=2
Ymin= ­ 20 and Ymax=20 with YScl=1
Sketch what you see below:
Jan 4­9:55 AM
8.1 sequences comp.notebook
January 10, 2017
Ex.14 Graphing using Sequence Graphing Mode
Graph the sequence defined recursively by:
b1=5
bn=bn­1+3 ∀n≥2
Use Seq mode
Use dot mode
Replace bn by u(n)
Parameters: ­ Select nMin=1, U(n)=u(n­1)+3, U(nMin)=5
Window ­ nMin=1, nMax=10, Plotstart=1, PlotStep=1 use the window [0,10] by [­5,25] graph.
Sketch what you see below:
Jan 4­10:14 AM