1.
Objectives:
1. Identify arithmetic and geometric
sequences
2. Find formulas for the nth term
2.
What is a Sequence?
A set of numbers, called
terms, arranged in a particular order.
Two simplest types:
Arithmetic
Geometric
3.
Arithmetic Sequences
Difference between consecutive
terms is constant.
Called the “common difference.”
Examples:
2, 6, 10, 14, 18, … diff. = 4
17, 10, 3, -4, -11, -18, … diff. = -7
a, a+d, a+2d, a+3d, a+4d, … diff. = d
4.
Geometric Sequences
Ratio of consecutive terms is
constant.
Called the “common ratio.”
Examples:
1, 3, 9, 27, 81, … ratio= 3
64, -32, 16, -8, 4, … ratio = -1/2
a, ar, ar2, ar3, ar4, … ratio = r
5.
You Try!
Identify the type of sequence and the
common difference or ratio.
5, 10, 20, 40, …
6, 1, -4, -9, …
6.
Notation
1st term: t1, 2nd term: t2, nth term: tn
Some sequences can be defined by
rules or formulas.
Ex: tn = n2 + 1
t1 = 12 + 1 = 2
t2 = 22 + 1 = 5, and so on
7.
Arithmetic Formulas
tn
nth term
=
t0
+
0th term
(work
backwards
to find)
dn
add the
difference
n times.
8.
Geometric Formulas
tn
nth term
=
t0
0th term
∙
rn
multiply by
the ratio
n times
9.
Formal Definition
“A function whose domain is the set
of positive integers.”
For example:
The sequence tn = 4n – 2
- can be thought of as The function t(n) = 4n – 2
(where n is a + integer)
10.
Graphing Sequences
Write terms as ordered pairs and plot.
Ex: 1, 4, 7, 10, …
has points (1, 1), (2, 4), (3, 7), (4, 10)
Notice n (the term number) is the x!
Arithmetic – points lie on a line
Geometric – points lie on an
exponential curve
11.
Example 1:
Find formula for nth term of 3, 5, 7…
Sketch the graph.
12.
Example 2:
Find formula for nth term of
3, 4.5, 6.75…
Sketch the graph.
13.
You Try!
Find formula for nth term of
15, 7, -1, -9, …
100, -50, 25, -12.5, …
14.
Example 3:
In a geometric sequence, t3 = 12
and t6 = 96. Find t11.
15.
Example 4:
In an arithmetic sequence t2 = 2 and
t5 = 16. Find t10.
16.
You Try!
In a geometric sequence t2 = 2 and
t5 = 16. Find t10.
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