Transcript Arithmetic Sequence

Arithmetic and
Geometric Sequences
by Pam Tobe
Beth Bos
Mary Lou Shelton
• Suppose you have $40 in a piggy
bank that you are saving to spend on
a special project. You take on a
part-time job that pays $13 per day.
Each day you put the cash into the
piggy bank. The number of dollars in
the bank is a function of the number
of days you have worked.
Days Dollars
n
tn
1
53
2
66
3
79
4
92
5
105
But it is a discrete
function rather than a
continuous function.
After 3 ½ days you still
have the same $79 as
you did after 3 days. A
function like this, whose
domain is the set of
positive integers, is
called a sequence.
Objectives:
• Represent sequences
explicitly and recursively
Given information about a sequence
• Find a term when given its term
number
• Find the term number of a given
term.
Let’s investigate the sequence of
dollars 53, 66, 79, 92, 105… in the
previous problem by:
a. Sketching the graph of the first few
terms of the sequence.
b. Finding t100n the 100th term of the
sequence
c. Writing an equation for tn the nth
term of the sequence, in terms of n.
a) The graph shows
discrete point. You
may connect the
points with a dashed
line to show the
pattern, but don’t
make it a solid line
because sequences
are defined on the
set of natural
numbers.
b. To get the fourth term, you add the
common difference of 13 three times
to 53. So to get the 100th term, you
add the common difference 99 times
to 53.
t100n = 53 + 99(13) = 1340
c.
tn = 53 +13(n-1)
• The sequence in Example 1 is called
an arithmetic sequence. You get
each term by adding the same
constant to the preceding term. You
can also say that the difference of
consecutive terms is a constant. The
constant is called the common
difference.
• The pattern “add 13 to the previous term
to get the next term” in Example 1 is
called a recursive pattern for the
sequence. You can write an algebraic
recursion formula
tn = tn-1 + 13
Sequence mode:
nMin = 1
beginning value of n
u(n) = u(n-1)+13 recursion formula
u(uMin) = {53}
enter first term
Press Graph
The pattern tn = 53 + 13(n-1) is called
an explicit formula for the
sequence. It “explains” how to
calculate any desired term without
finding the terms before it.
Arithmetic Sequence:
A sequence in which consecutive terms differ by a
fixed amount is an arithmetic sequence, or arithmetic
progression.
Definition: Arithmetic Sequence
A sequence (an) is an arithmetic
sequence (or arithmetic progression) if it
can be written in the form:
an = an-1 + d
n> 2
For some constant d. The number d is
the common difference.
Pair/Share, Try It
Determine whether the sequence
could be arithmetic. If so, find the
common difference.
a) -6, -3.5, -1, 1.5, 4,……
b) 48, 24, 12, 6, 3,…..
c) In3, In 5, In12, In24
If (an ) is an arithmetic sequence with
common difference d, then
a 2 = a1 + d
a3 = a2 + d = a1 + 2d
a4 = a3 + d = a1 + 3d
nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence can be
written in the form: an = a1 + (n – 1)d
Where a1 is the first term and d is the common
difference
Pair/Share, Try It
The third and eighth terms of an
arithmetic sequence are 13 and 3,
respectively. Find the first term, the
common difference, and an explicit
rule for the nth term.
Geometric Sequence
In an arithmetic sequence, terms are
found by adding a constant to the
preceding term. A sequence in
which terms are found by multiplying
the preceding term by a (nonzero)
constant is a geometric sequence or
geometric progression.
Definition: Geometric Sequence
A sequence (an) is a geometric sequence (or
geometric progression) if it can be written in the
form
an = an-1 * r,
n>2
Where r ≠ 0 is the common ratio.
Pair/Share, Try It
Determine whether the sequence could
be geometric. If so, find the common
ratio.
2 2 2 2
a) 2, , ,
,
.........
3 9 15 21
b) 3, 6, 12, 24, 48 .......
c) 10-3, 10-1, 101, 103, 105 .......
nth Term of a Geometric Sequence
The nth term of a geometric sequence can
be written in the form.
an = a1 * rn-1
Pair /Share, Try It
The third and eighth terms of a geometric
sequence are 20 and -640, respectively.
Find the first term, common ratio, and an
explicit rule for the nth term.
Application
• The population of Bridgetown is growing at
the rate of 2.5% per year. The present
population is 50,000. Find a sequence that
represents Bridgetown’s population each
year. Represent the nth term of the
sequence both explicitly and recursively.
Evaluate seven terms of the sequence.
State
New York
Texas
Population
(1993)
Growth rate
18,197,000
18,031,000
0.5%
2.0%
Assume that the population of New York and
Texas continued to grow at the annual
rate as shown:
a) In what year will the population of Texas
surpass that of New York?
b) In what year will the population of Texas
surpass that of New York by 1 million?