P.o.D.
3⁄
256 4
1.) Evaluate
3
2.) Simplify √64𝑥 3 𝑦 6 𝑧 with rational
exponents.
3.) Solve 9 = 3𝑥 − 2
4.) Solve log 𝑥 243 = 5
5.) Solve 3.4𝑥 = 82.3
1.) 64
2 1⁄3
4𝑥𝑦 𝑧
2.)
3.) 2.1827
4.) 3
5.) 3.6039
9-1: Sequences and Series
Learning Target: I will be able to use
sequence notation to write the terms of
sequences; use summation notation to
write sums; find the sums of infinite
series; use sequences and series to
model real-life problems.
Definition of Sequence:
An infinite sequence is a function whose
domain is the set of positive integers.
The function values 𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛 , …
are the terms of the sequence. If the
domain of the function consists of the
first n positive integers only, the
sequence is finite.
EX: Write the first five terms of each
sequence.
a.) 𝑎𝑛 = 2𝑛 + 1
𝑎1 = 2(1) + 1 = 3
𝑎2 = 2(2) + 1 = 5
𝑎3 = 2(3) + 1 = 7
𝑎4 = 9, 𝑎5 = 11
b.) 𝑎𝑛 = 2 − (−1)𝑛
3,1,3,1,3
EX: Write the first five terms of 𝑎𝑛 =
2+(−1)𝑛
𝑛
1, 3/2, 1/3, ¾, 1/5
EX: Write an expression for the nth term
of each sequence.
a.) 1,5,9,13,17
Notice that each term is 4 more than the
previous.
We need to think of a formula for each
term in the sequence that has a factor of
4. In other words, 4n must be in the
equation.
𝑎𝑛 = 4𝑛 − 3
b.) 1,3,5,7
𝑎𝑛 = 2𝑛 − 1
Some sequences are defined recursively.
In other words, you need the previous
term to find the next term.
EX: Find the first five terms of the
sequence defined recursively by
𝑎1 = 6
{
𝑎𝑘+1 = 𝑎𝑘 + 4
6, 10, 14, 18, 22
Explicit: 4n+2
The Fibonacci Sequence is an example of
a recursive formula.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
𝑎0 = 1
𝑎1 = 1
{
𝑎𝑘 = 𝑎𝑘−2 + 𝑎𝑘−1
Each term is found by adding the
previous two terms.
Now take each term in the Fibonacci
Sequence and divide it by the previous
term…
This is known as the Golden Ratio, 𝜑 =
1+√5
2
≈ 1.618. It is the most common
number in nature.
http://www.youtube.com/watch?v=03uC9bhX0Rc
http://www.youtube.com/watch?v=oT_Bxgah9zc
Factorial Notation:
If n is a positive integer, n factorial is
defined as n!=1(2)(3)(4)(5)(…)(…)(n-1)(n).
As a special case, zero factorial is defined
as 0!=1.
EX: Write the first 5 terms of the
sequence given by 𝑎𝑛 =
3𝑛 +1
𝑛!
. Begin with
n=0. Then graph the terms on a set of
coordinate axes.
30 + 1 1 + 1 2
𝑎0 =
=
= =2
0!
1
1
31 + 1 3 + 1 4
𝑎1 =
=
= =4
1!
1
1
32 + 1 9 + 1 10
𝑎2 =
=
=
=5
2!
2
2
14
41
𝑎3 =
, 𝑎4 =
3
12
Now plot the points (0,2), (1,4), (2,5), (3,
14/3), (4,41/12).
EX: Evaluate each factorial expression.
a.)
a.)
b.)
c.)
9!
b.)
3!7!
362880
30240
241920
3!8!
4!4!
c.)
(𝑛+1)!
𝑛!
= 12
= 420
576
(𝑛+1)(𝑛)(𝑛−1)(𝑛−2)(𝑛−3)…
(𝑛)(𝑛−1)(𝑛−2)(𝑛−3)…
=𝑛+1
Summation Notation (Sigma Notation):
The sum of the first n terms of a
sequence is represented by ∑𝑛𝑖=1 𝑎𝑖 =
𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛 where i is called
the index of summation, n is the upper
limit of summation, and 1 is the lower
limit of summation.
EX: Find each sum.
a.) ∑4𝑖=1(4𝑖 + 1)
={4(1)+1}+{4(2)+1}+{4(3)+1}+{4(4)+1}=
5+9+13+17=44
b.) ∑5𝑘=2(2 + 𝑘 3 )
= ( 2 + 2 3 ) + ( 2 + 33 ) + ( 2 + 4 3 )
+ (2 + 53 )
= 10 + 29 + 66 + 127 = 232
c.)
6 2
∑𝑖=1
𝑖!
2 2 2 2 2 2
= + + + + +
1! 2! 3! 4! 5! 6!
1 1
1
1
=2+1+ +
+
+
3 12 60 360
1237
=
≈ 3.4361
360
*There is a way to perform summation
(sigma) notation on your calculator.
Series = sum of a sequence
EX: For the series ∑∞
𝑛=1
5
10𝑛
, find (a) the
third partial sum and (b) the sum.
a.)
5
3
∑𝑛=1 𝑛
10
5
5
5
111
= 1 + 2 + 3 = 0.555 =
10
10
10
200
b.) We are trying to find the sum of an
infinite sequence. What is
happening to each successive term?
Notice that the terms are
approaching 0. Essentially, we are
adding infinitesimally small values,
or nothing at all. We can trick the
calculator into solving this for us.
EX: The average cost to community
hospitals per patient per stay from 1997
to 2004 is approximated by the model
𝑎𝑛 = 4.121𝑛3 − 89.65𝑛2 + 615.3𝑛 + 4907,
𝑛 = 0,1,2, … ,7, where 𝑎𝑛
is the cost in
dollars and n represents the year with
n=0 corresponding to 1997. Find the last
three terms of this sequence, which
represent the cost to community
hospitals per patient per stay from 2002
to 2004.
2002 to 2004 has n=5, 6, 7.
Upon completion of this lesson, you
should be able to:
1. Write the terms of an arithmetic
sequence.
2. Write an expression for a sequence.
3. Write both explicit and recursive
formulas for sequences.
4. Evaluate expressions in factorial
notation.
5. Evaluate sigma (summation)
notation.
For more information, visit
http://www.purplemath.com/modules/series.htm
HW
Pg.649
6-96 6ths, 125-128