Download 9.4 Sequences sequence: ordered progression of numbers Find the

May 07, 2010
9.4 Sequences
sequence: ordered progression of numbers
Find the 100th term of the sequence
7, 10, 13, 16, ...
May 07, 2010
Examples:
1.
4, 8, 16, 20, 24
2.
3, 9, 27, 81, 243, ..., 3k, ...
3.
{a1, a2, a3, ..., ak, ...}
finite:
infinite:
May 07, 2010
Explicitly defined functions define them as a function of k.
Ex. Find the first 6 terms and the 100th term of the
sequence {ak} in which ak = k2-1.
May 07, 2010
Recursively defined sequences uses a formula relating
each term to previous terms.
Example: Find the first 6 terms and the 100th term for
the sequence defined recursively by the conditions:
b1 = 3
b2 = bn-1 + 2
for all n > 1.
May 07, 2010
Limits of Infinite Sequences:
(What is its end behavior?)
Let {an} be a sequence of real numbers, and consider lim an.
If the limit is a finite number, L, the sequences converges and
L is the limit of the sequence.
If the limit is infinite or nonexistent, the sequence diverges.
May 07, 2010
Determine whether the sequence converges or diverges. If
it converges, give the limit.
1.
1/2, 1/4, 1/6, 1/8, ..., 1/(2n), ...
2.
11/1, 12/2, 13/3, 14/4, ...
3.
5, 10, 15, 20, 25, ...
4.
0.1, 0.2, 0.3, 0.4, 0.5, ...
5.
-1, 1, -1, 1, -1, 1, ... (-1)n, ...
May 07, 2010
Determine whether the sequence converges or diverges. If it
converges, give the limit.
1.
3.
3n
n+1
3
n +2
n2 + n
2.
5n2
n3+1
May 07, 2010
Arithmetic Sequence:
has the form: {a, a + d, a + 2d, a + 3d, ... a + (n-1)d, ...},
for some constant, d (called the common difference).
a =a
+d
n
n-1
n
1
2
3
4
n
Term
a1
(for all n > 2)
May 07, 2010
For each of the following arithmetic sequences, find
a. the common difference
b. the tenth term
c. a recursive rule for the nth term
d. an explicit rule for the nth term
1.
2.
5, 8, 11, 14, 17, ...
ln 3, ln 9, ln 27, ln 81, ...
May 07, 2010
Geometric Sequence:
has the form {a, ar, ar2, ar3, ... arn-1, ...} for some
nonzero constant r (common ratio).
an = an-1 r
n
1
2
3
4
n
Value
a1
(for all n > 2)
May 07, 2010
For each of the following geometric sequences, find
a. the common ratio
b. the tenth term
c. a recursive rule for the nth term
d. an explicit rule for the nth term
1.
2, 6, 18, 54, 162, ...
2.
26, 29, 212, 215, 218, ...
May 07, 2010
Constructing Sequences
1. The second and fifth terms of a sequence are 3 and 192,
respectively. Find explicit and recursive formulas for the
sequence if it is
a. arithmetic
b. geometric
May 07, 2010
The Fibonacci Sequence:
a1 = 1
a =1
2
an = an-2 + an-1
(for all positive integers n > 3)