9.1 NOTES
Sequences and Series
A Sequence in math is a collection of
terms or numbers that have a specific order
A Series in math is the sum of the terms
of a finite (or infinite) sequence
Pre Calc 9.1 Sequences and Series
A SEQUENCE is a collection of terms or numbers thathave a specific order
A SERIES in math is the sum of the terms of a finite or infinite
sequence.
First 5 Greek Letters:
SUMMATION NOTATION OR SIGMA NOTATION alpha
beta
Uses the Greek letter Sigma, written as
gamma
delta
Epsilon
α
β
γ
δ
ε
Σ
Α
Β
Γ
Δ
Ε
EX1 Write the first 4 terms of each sequence:
1. an = 3n ­ 2
a1= 1 =
a2= 2 =
a3= 3 =
a4= 4
=
2. an = 3 + (­1)n
1
a1=
=
2
a2=
=
3
a3=
=
4
a4=
=
4. an = n(n ­ 2) 5. an = 12
a1= 1(1­2) =
a1=
a2=
a2= 2(2­2) =
a3=
a3= 3(3­2) =
a4=
a4= 4(4­2) =
subscript numbers
3. an = (­1)n
2n ­ 1
a 1=
1
=
1
2
a 2=
=
2
3
a 3=
3
=
a 4=
4
4
=
EX2
Write the first 5 terms of the sequence defined recursively:
"recursively" means a recurrent or repeating pattern
1. a1 = 5, a k+1 = ak + 4
next term in last term add 4 to last
the sequence you found term to get
new term
Answer
2. a1 = 8, a k+1 = ­2(ak ­ 1)
next term in
the sequence
Answer
last term
you found
a 1= 5
a 2= + 4 =
a 3= + 4 =
a 4= + 4 =
a 5= + 4 =
a 1= 8
a2= ­2(
a3= ­2(
a4= ­2(
a5= ­2(
­1) =
­1) =
­1) =
­1) =
FACTORIAL!
CALCULATOR KEYSUSED
TI­84
TI­Nspire
! MATH PRB ?!>
n Factorial is defined as
n! = n
(n­1)
!
lower right corner of calculator
5 4 3 2 1
0! = 1
Find each:
1. 0! =
2. 1! =
3. 2! =
4. 3! =
5. 4! =
6. 5! =
=
=
=
=
7. 9!
987654321
7! = 7 6 5 4 3 2 1 =
8. 6!
654321
=
=
5! 3!
54321 321
=
9. 3! 8! 3 2 1 8 7 6 5 4 3 2 1
4! 6! = 4 3 2 1 6 5 4 3 2 1 =
=
10. 12! 12 11 10 9 8 7 6 5 4 3 2 1
11! = 11 10 9 8 7 6 5 4 3 2 1 =
50
100
=
=
3
6
Sequence of numbers
A sequence of numbers is a collection of terms or numbers
that have a specific order
Below are examples of what the terms look like to the left
and to the right of "n" after "n" is defined.
the term "n" will need to be defined within the problem so that you
know where to begin
EX4
Write the first five terms of the sequence.
Assume that n begins with 0.
2
1. an = n!
a =
0
a =
1
a =
2
a =
3
a =
4
Answer
2, 2, 1, 1 , 1
12
2. an = n ‐1
a =
a =
a =
a =
a =
Answer
­1, 0, 1, 2, 3
Using a Calculator to find a sequence of numbers:
( type in information below )
(formula, variable, lower limit, upper limit)
In calculator mode: Select seq from the Catalogue or simply type in the letters seq and then finish typing in the rest of the information
Find: seq(3x2,x,1,8)
= 3 12 27 48 75 108 147 192
= {3, 12, 27, 48, 75, 108, 147, 192}
this is what calc
will show but this
is not how you
write the answer
Answer
Using a Calculator to find a summation of numbers:
Find: sum(seq(2x,x,1,6)
=
108
Answer
SUMMATION NOTATION
= Sigma
Σ The sum of the first n terms of a sequence is represented by
Where i is called the INDEX of summation
n is called the UPPER LIMIT of summation
1 is called the LOWER LIMIT of summation
EX 5 Find Each Algebraically:
Expansion/Work
Answer
Using a Calculator to find summation:
TI­84
SUM LIST OPS SEQ
LIST MATH
SUM(SEQ( formula, variable , lower , upper , increment )
limit limit
used
this part can be omitted
TI­Nspire
In calculator mode: Select sum and seq from the Catalogue "increment" is optional and not needed or simply type sum and seq unless you are jumping numbers
SUM(SEQ(formula, variable , lower , upper , increment )
limit limit this part can used
be omitted
x
Try this: sum(seq(2 ­3, x,1,6)) = 108
Ex 6 Use a calculator to find each:
45
90
8
3
76
124
Homework
pg 649:1­11odd, 19, 23, 25, 53, 62, 65­71odd
WPF = Answers only (for all)
73­83 odd WPF, show expansion of numbers,
then answer
85­88 all WPF answer (use calculator to find
the answer)
99, 100, 102 WPF, show expansion of numbers,
then answer
Answers to the Even problems
62.
86. 55991 or 6.06
9240
88. 3
8
100.
102.