Transcript 13.1 Arithmetic and Geometric Sequences

13.1 Arithmetic and
Geometric Sequences
Explicit Formulas
13.1 Arithmetic and Geometric
Sequences
Objectives:
Identify Arithmetic and Geometric
sequences
2. Complete missing information
associated with sequences
3. Define formulas for sequences
1.
Vocabulary:
Sequence, term, Arithmetic, Geometric
Learning The Lingo

A "sequence" an ordered list of numbers;
the numbers in this ordered list are called
"elements" or "terms".
Learning the Lingo

A sequence may be named or referred to as "A"
or "An".
The terms of a sequence are usually named
something like "ai" or "an", with the subscripted
letter "i" or "n" being the "index" or counter.

An =
a1
a2
a3
a4
a5
Arithmetic Sequences



The two simplest sequences to work with are
arithmetic and geometric sequences.
An arithmetic sequence goes from one term to
the next by always adding (or subtracting) the
same value.
For instance:
Add 3 to each term
 Example
One: 2, 5, 8, 11, 14,...
 Example Two: 7, 3, –1, –5,...
Subtract 4 to each term
Learning The Lingo

The number added (or subtracted) at each
stage of an arithmetic sequence is called
the "common difference" d, because if you
subtract (find the difference of) successive
terms, you'll always get this common
value.
Writing Arithmetic Sequences
an = a1 + (n – 1)d
Example – Your Turn
Find the common difference and
the next term of the following
sequence: 3, 11, 19, 27, 35,...
The difference is always 8, so d = 8.
Then the next term is 35 + 8 = 43.
Examples: Find a formula for an
and find the 10th term

2,6,10,14,18,…

17,10,3,-4,-11,-18,…
Examples:

Find the n-th term (formula) of the
arithmetic sequence having a4 = 93 and
a8 = 65.
Since a4 and a8 are four places apart, then I
know from the definition of an arithmetic
sequence that a8 = a4 + 4d.
65 = 93 + 4d
–28 = 4d
–7 = d
93 = a + 3(–7)
93 + 21 = a
114 = a
OR
65 = a + 7(–7)
65 + 49 = a
114 = a
Geometric Sequences
A geometric sequence goes from one term
to the next by always multiplying (or
dividing) by the same value.
 For instance:

 Example
One: 1, 2, 4, 8, 16,... Multiply by 2 at each step
 Example Two: 81, 27, 9, 3, 1, 1/3,...
Divide by 3 at each step
Learning The Lingo

The number multiplied (or divided) at each stage
of a geometric sequence is called the "common
ratio" r, because if you divide (find the ratio of)
successive terms, you'll always get this common
value.
Writing Geometric Sequences
an = a1
(n
–
1)
r
Example – Your Turn!
Find the common ratio and the
seventh term of the following
sequence:
2/9, 2/3, 2, 6, 18,...
The ratio is always 3, so r = 3.
Then the sixth term is (18)(3) = 54 and the
seventh term is (54)(3) =162
Examples: Find a formula for an
and find the 10th term

1,3,9,27,81,…

64,-32,16,-8,4,…
Example

Find the n-th (formula) of the geometric
sequence with a5 = 5/4 and a12 = 160.
These two terms are 12 – 5 = 7 places apart, so, from the
definition of a geometric sequence, I know that
7
160 = (5/4)(r )
128 = r7
2=r
4
5/4 = a(2 ) = 16a
5/64 = a
a12=a5r7
11
160 = a(2 ) = 2048a
OR
160/2048 =5/64= a
Homework:

Textbook: p. 477 #17-21, 29, 33
Practice

WB p. 91 # 4
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No, multiply is Geometric
Practice

WB p. 98 # 3

an=3(1/4)(n-1) or an=12(1/4)(n)

a8=3/47 or a8=0.0001831
Practice

WB p. 91 # 8

an= 1 + (n – 1)-5 or an= - 5n + 6

a30= -144
Practice

WB p. 97 # 3
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an=1024(+/-1/4)(n-1)
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an=4096(+/-1/4)(n)
Practice

WB p. 91 # 12

an= -81 + (n – 1) 11 or 11n + 92