Chapters 2 and 9 Sequences and Series and the Binomial Theorem
Assignments
2A
2B
2C
2D
2E.1
2E.2
2E.3
2F
9A
9B
Page
Page
Page
Page
Page
Page
Page
Page
Page
Page
36:
40:
42:
46:
51:
52:
54:
57:
197:
200:
2, 4, 5
1dg, 2, 3
2, 4, 5ace, 6ac, 7b, 8, 9
1, 3, 5, 6ab, 7ab, 8ac, 10, 18, 20
1adf
1bc, 2b, 4, 6, 7ab, 9, 11
1a, 2ad, 4, 6, 7(4d)
1ac, 3ac, 4ac, 5a
1beh, 2bdf, 3cd,4abii, 5ac, 6, 7, 8a
1c, 2ac, 3ab, 4a
Review Sets
2A Page
2B Page
2C Page
9
Page
59:
59:
60:
202:
1bc, 2-9
2, 5-9
1, 2, 4, 6, 7
1-7
Sequences and Series
2A – Number Patterns
Identify the pattern to find the next three numbers: (Assume that the pattern continues.)
1) 1, 4, 9, 16, 25, _____, _____, _____, …
2) 60, 54, 66, 60, 72, _____, _____, _____, …
3) -1, 3, -9, 27, -81, _____, _____, _____, …
4) 1, 3, 6, 11, 18, 29, _____, _____, _____, …
Definition of a Sequence
Sequences are ordered lists of numbers.
A finite sequence is a function whose domain consists of the m integers 1, 2,
3, …, m. An infinite sequence’s domain is the set of all positive integers.
The numbers in the range (or the numbers in the ordered list) are called
terms. The notation a1 , a 2 ,... is used and, in general, a n is the nth term of the
sequence.
2B – Sequences of Numbers
So, a n  5n  6 represents the sequence –1, 4, 9, 14, 19, ….. This formula (general term or nth
term) is the rule for the sequence and is given explicitly, which means that any term in the
n
sequence may be found directly from the formula. Note: The factor  1 in a general term causes
the terms to alternate between _________________ and _________________.
Write the first four terms of the sequence whose nth term, or general term, is given:
A.
an  3  1
n
B.
a n  100  n
2
C.
an
n

 1

2n  1
2C - Arithmetic Sequences
Arithmetic Sequence – a sequence in which a common difference exists between any two
consecutive terms of the sequence. If this difference is d  a n  a n 1 , the second part of the
recursive formula is given by a n  a n 1  d . The explicit formula (general term or nth term) of an
arithmetic sequence is a n  a1  (n  1)d .
How does this relate to the equation of a line?

Find the Common Difference of an Arithmetic Sequence
A.
7, 4, 1, -2, -5, … is an arithmetic sequence. Find the common difference:
B.
-17, -13, -9, -5,… is an arithmetic sequence. Find the common difference:

Write Terms of an Arithmetic Sequence
The recursion formula an  an1  2.18 models the percentage of women working in the U.S.
labor force, a n , for each five-year period starting with 1980. In 1980, 51.5% of U.S. women
were working in the labor force. Find the first five terms of the arithmetic sequence in which
a1  51.5 and an  an1  2.18 .

Use the Formula for the General Term of an Arithmetic Sequence
A.
Find the ninth term of the arithmetic sequence whose first term is 6 and whose common
difference is -5.
B.
According to the U.S. Census Bureau, new one-family houses sold for an average of
$159,000 in 1995. This average sales price has increased by approximately $9700 per year.
a. Write a formula for the nth term (general term) of the arithmetic sequence that describes
the average cost of new one-family houses n years after 1994.
b. How much will new one-family houses cost, on average, by the year 2010?
k
given that
3k  1, k and  3 are consecutive terms of an arithmetic sequence.
C.
Find
D.
Find the general term
E.
Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence.
un
for an arithmetic sequence given that u 3  8 and
u8  17 .
2D - Geometric Sequences
Geometric Sequence – a sequence in which a common ratio exists between any two consecutive
terms of the sequence. If this ratio is r  a n  a n 1 , the second part of the recursive formula is
given by a n  a n 1  r . The explicit formula (general or nth term) of a geometric sequence is
a n  a1 r n 1 .

Find the Common Ratio of a Geometric Sequence
A.
8, 4, 2, 1, ½, … is a geometric sequence. Find the common ratio:
B.
1, -3, 9, -27, 81,… is a geometric sequence. Find the common ratio:

Write Terms of a Geometric Sequence
Write the first six terms of the geometric sequence with first term 12 and common ratio

1
2
:
Use the Formula for the General Term of a Geometric Sequence
A.
Find the seventh term of the geometric sequence whose first term is 5 and whose common
ratio is -3.
B.
Write the general term for the geometric sequence 3, 6, 12, 24, 48,… Then use the formula
for the general term to find the eighth term.
C.
Find
D.
A geometric sequence has u2  6 and
k
given that
k  1, 2k and 21  k are consecutive terms of a geometric sequence.
u5  162 .
Find its general term.

Compound Growth
a n 1  a1r n is used for compound growth where u1 is the initial investment, r is the growth
multiplier, n is the number of years, and u n 1 is the amount after n years.
A.
$12,000 is invested for 5 years at 6% p.a. compound interest. What will it amount to at the
end of this period?
B.
How much should I invest now if I want the maturing value to be $20,000 in 5 years’ time, if
I am able to invest 7.5% p.a. compounded annually?
2E and F - Series and Sigma Notation

Use Summation or Sigma Notation
Summation or Sigma Notation
A series is the sum of the terms of a sequence. They may be finite or infinite,
like a sequence.
Sigma or Summation Notation uses the Greek letter Sigma (S for sum) to
write the series:
a  a  a  a      a
n
i
1
2
3
n
.
i 1
So, this represents the sum of ai from i=1 to i=n. i is called the index of the
summation. 1 (this can vary) is the lower limit of summation and n is the
upper limit of summation. We use the infinity symbol for n in an infinite series.
Properties:
1.
2.
n
n
i 1
i 1
 cai  c ai
n
n
n
i 1
i 1
i 1
 ai  bi    ai   bi
Expand and evaluate the sum:
A.
6
 2i 2
B.
i 1
 2
5
k

3
C.
k 3

a  ar  ar 2      ar 14
4
i 1
Express each sum using summation notation:
12  2 2  32      9 2
1  12  14  18     
A.
B.
D.
5
E.
1
2 n 1
C.
6  8  10      32
a  d   a  d 2      a  d n 
Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence
Series – the sum of the terms of a sequence. They may be finite or infinite, like a sequence.
Sigma or Summation Notation – Uses the Greek letter Sigma (S for sum) to write the series:
n
a
k 1
k
 a1  a 2  a 3      a n . So, this represents the sum of a k from k=1 to k=n.
k is called the index of the summation. We use the infinity symbol for n in an infinite series.
Express each series using sigma notation:
A.
5 + 10 + 15 + 20 + 25 + 30 + 35
B.
-3 – 6 – 9 – 12 – 15 – 18 - …
The sum of the first n terms of a series is called the nth partial sum and is denoted Sn. For an
arithmetic sequence, the sum is given by S n  n2 a1  an  .
Find the indicated partial sum:
C.
15
 5k  3
1
E.
D.
20
 3m  10
10
25
 6i  11
i 1
F.
Find the sum of the first 15 terms of the arithmetic sequence: 3, 6, 9, 12, …
G.
Your grandmother has assets of $350,000. One option that she is considering involves
nursing home care for a ten-year period beginning in 2001. The model an  1800n  49,730
describes yearly nursing home costs n years after 2000. How much would it cost for 10
years? Will she be able to afford it?
Redo the following problems using your graphing calculator and the “Sum” and “Seq” capabilities:
H.
20
 3m  10
I.
 6i  11
i 1
10

25
Use the Formula for the Sum of the First n Terms of a Geometric Sequence
Express each series using sigma notation:
A.
8, 4, 2, 1, ½, …
B.
1, -3, 9, -27, 81
The sum of the first n terms of a geometric sequence is given by S n 


a1 1  r n
,r 1 .
1 r
Find the indicated partial sum:
C.
10
5 2 j
D.
8
23
i 1
 3  0 .9
k
k 1
j 1
E.
4
F.
S7 for –343 + 49 – 7 + 1 - …
i 1
G.
A job pays a salary of $30,000 the first year. During the next 29 years, the salary increases
by 6% each year. What is the total lifetime salary over the 30-year period?
Redo the following problems using your graphing calculator and the “Sum” and “Seq” capabilities:
H.
10
5 2 j
I.
 3  0 .9
k
k 1
j 1

4
Use the Formula for the Sum of an Infinite Geometric Series
The Sum of an Infinite Geometric Series
If r  1,
S
a1
. The sum exists and we say that the series converges.
1 r
If r  1 , the infinite series does not have a sum. We say that it diverges.
A.
Find the sum of the infinite geometric series: 3  2  43  89    
B.
Express 0. 9 and 0.83 as fractions in lowest terms.
C.
A tax rebate that returns a certain amount of money to taxpayers can have a total effect on
the economy that is many times this amount. In economics, this is called the multiplier
effect. The government reduces taxes so that each consumer has $1000 and each person
spends 80% of this, and the individuals and businesses receiving this in turn spend 80% of
it, and so on. Determine the total amount spent on consumer goods from the initial $1000
rebate.

Use Factorial Notation
Factorial Notation
If n is a positive integer, the notation n! is the product of all positive integers
from n down to 1.
n! n(n  1)( n  2)    3  2 1
0! 1 , by definition.
Evaluate each factorial expression:
A.
14!
2! 12!
B.
n!
(n  1)!
The Binomial Theorem
9A and 9B - Binomial Expansions
A.
Expand the following: (Write answers in standard form).
B.
List the coefficients, in order, of each of your polynomials above:
a  b 0 
a  b 1 
a  b 2 
a  b 3 
a  b 0
a  b 1
a  b 2
a  b 3
What do you notice about the pattern of the coefficients?
Notice the pattern of the exponents for a, b, and for the sum of a and b.
How many terms are there in the polynomial compared to the power of the binomial?
Predict what the expanded form of a  b  would be.
4

Evaluate a Binomial Coefficient
Elements of Pascal’s Triangle, and binomial coefficients, can also be found using the formula:
n
n!
  n C r 
r!n  r !
r
Evaluate the following:
A.
6
 
 3
B.
6
 
0
C.
8
 
 2
D.
 3
 
 3
E.
10 
 
3
F.
 20 
 
 15 

Expand a Binomial Raised to a Power
The Binomial Theorem
For any positive integer n and any numbers a and b,
a  bn  a n  
n  n1  n  n2 2
 n  2 n2  n  n1
a b   a b      
a b  
ab  b n
1
 2
 n  2
 n  1
or
a  b n   
n  nr r
a b
r 0  r 
n
Expand using the Binomial Theorem:
x  14
A.
where
n
n!
  n C r 
r!n  r !
r
B.
4a  b 5
C.
x  2 y 5
D.
x
E.
3x  56
F.
1

 3x  
x

G.
3

x 
x

H.
3  3 
2
 2y

5
4
4
4
I.
3  x 5
Use the above to find the value of 3.1 .
5

Find a Particular Term in a Binomial Expansion
Finding a Particular Term in a Binomial Expansion
n
The r  1 st term of the expansion of a  b  is  a n  r b r .
r
n
Find the indicated term:
11
A.
The term containing x 3 in  x  y 
B.
The term containing y 3 in  x  y 
12
C.
The constant term of 2 x  7 
D.
The constant term of  x  3
E.
The 6th term of a  12 
F.
The 3rd term of 2 x  y 
G.
The 4th term of 2 x  y 
H.
The 5th term of x 2  3x
I.
2

The 8 term of  x 3   .
x

4
9
9
15
th
5
12


8


4 

x3 
12
J.
The constant term in the expansion of  x 
K.
The coefficient of a 4 b 2 in the expansion of a  3b  .
L.
If f ( x)  x 5 , find

A.
B.
6
f x  h   f ( x)
and simplify.
h
Pascal’s Triangle
Create Pascal’s Triangle out to 6 rows:
n
We name the nth row and rth element by using the notation   . But don’t start counting
r
at 1. Instead, start counting rows and elements at __________.
 
So, the first 6 that appears in Pascal’s triangle would be   .
 
C.
Cool facts about Pascal’s Triangle:
a) If the 1st element in a row is prime, then all of the other elements in that row except the
1’s are ________________ by that 1st element.
b) Hocky Stick Pattern: Choose a diagonal of any length starting with an outer 1. The
_________ of any number of the numbers in that diagonal will be found at the end
of that list on the opposite diagonal.
c) Magic 11’s: If a row is made into a single number using the elements as digits (carrying
over for multiple-digit elements), then that number equals _____________.
d) Triangular Numbers: The Triangular Numbers appear in the ______________ diagonal.
e) Perfect Squares: The Perfect Squares are generated by adding consecutive terms of the
list of ________________ _______________.