Download Lesson 9.1 Notes

9.1 Sequences Objec(ves: • Define sequence • Arithme(c and geometric sequences • Graph a sequence and determine the limit. Assignment: AB Book pg. 604 #’s 1-­‐33 odd, 35-­‐44 all Sequences
A sequence {an } is a list of numbers written in
explicit order.
an = {a1 ,a2 ,a3 ,...,an ,...}
a1 ! First term
a2 ! 2 nd term
a3 ! 3rd term
an ! n th term
Sequences
Any real-valued function with domain a subset of the
set of positive integers is considered a sequence.
If the domain is finite, then the sequence is a finite
sequence.
1 Example: Find the first 4 terms and the 100th term of
the sequence{an }where
n
an =
( !1)
n2 + 1
This sequence is an explicit
sequence because an is defined
in terms of n.
Another way to define a sequence is recursively
by giving a formula for an in terms of the
previous term.
Example: Find the first 4 terms and the 8th term for the
sequence defined recursively by the conditions
b1 = 4
bn = bn!1 + 2 for all n " 2
There are many types of sequences but two
particular types of sequences are dominant in
mathematical applications.
•  Arithmetic Sequences- pairs of successive
terms all have common differences.
•  Geometric Sequences- pairs of successive
terms all have common ratios.
2 Arithmetic Sequences
A sequence {an }is a arithmetic sequence if it can
be written in the form
{a, a + d, a + 2d,... a + (n ! 1)d,...}
for some constant d. The number d is the common
difference.
Each term in an arithmetic sequence can be
obtained recursively from its preceding term by
adding d:
an = an!1 + d, for all n " 2
Example: For the sequence, find the common
difference, the ninth term, a recursive rule for the nth
term and an explicit rule for the nth term.
-5, -2, 1, 4, 7, …
Common Difference:
Ninth Term:
Recursive Rule:
Explicit Rule:
Geometric Sequence
A sequence {an }is a geometric sequence if it can
be written in the form
{a, a ! r, a ! r ,... a ! r
2
n"1
}
,...
for some nonzero constant r. The number r is the
common ratio.
Each term in an geometric sequence can be
obtained recursively from its preceding term by
multiplying by r:
an = an!1 " r for all n # 2
3 Example: For the sequence, find the common ratio, the
10th term, a recursive rule for the nth term and an
explicit rule for the nth term.
1, -2, 4, -8, 16, …
Common Ratio:
10th Term:
Recursive Rule:
Explicit Rule:
Example: The second and fifth term of geometric
sequence are 6 and -48 respectively. Find the first
term, common ratio, and an explicit rule for the nth
term.
The main focus of this chapter concerns sequences whose terms approach limi(ng values. Such sequences are said to converge. 1 1 1 1 1
1
, , , , ,... n
2 4 8 16 32 2
This sequence converges to 0. 4 Limit
Let L be a real number. The sequence {an }has
limit L as n approaches ∞ if, given any positive
number ε, there is a positive number M such that
for all n > M we have
an ! L < !
We write lim an = L and say that the sequence
n!"
converges to L. Sequences that do not have limits
diverge.
There are different ways in which a sequences can fail to have a limit. One way is that the terms of the sequence increase without bound or decrease without bound. These cases are wriRen symbolically as follows: Properties of Limits
If L and M are real numbers and lim an = L and
n!"
lim bn = M , then
n!"
( an ± bn ) = L ± M
1.  Sum and Difference Rule lim
n!"
( anbn ) = L # M
2.  Product Rule lim
n!"
( c # an ) = c # L
3.  Constant Multiple Rule lim
n!"
4.  Quotient Rule
#a & L
lim % n ( = , M ) 0
n!" $ b '
M
n
5 Example: Determine whether the sequence
converges or diverges. If it converges, find its limit.
an =
2n ! 1
n
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