CHAPTER 7: SEQUENCES, SERIES, AND COMBINATORICS
7.1
SEQUENCES AND SERIES

SEQUENCE: A sequence is a function where the domain is a set of
consecutive positive integers beginning with 1.
o INFINITE SEQUENCE: An infinite sequence is a function
having for its domain the set of positive integers, 1, 2,3, 4,5,

o FINITE SEQUENCE: A finite sequence is a function having for
its domain a set of positive integers, 1, 2,3, 4,5, , n , for some
positive integer n.
o The function values are considered the terms of the sequence.
 The first term of the sequence is denoted with a
subscript of 1, for example, a1 , and the general term has
a subscript of n, for example, an .
o Example: Find the first four terms, a10 and a15 from the given
nth term of the sequence, an  n 2  1, n  3.
Solution:
The first four terms:
a3  32  1  8
a4  42  1  15
a10  102  1  99
a5  52  1  24
a15  152  1  224
a6  62  1  35
7,15, 24,35

Finding the General Term: When only the first few terms of a
sequence are known, we can often make a prediction of what the
general term is by looking for a pattern.
o Example: Predict the general term of the sequence -1, 3, -9,
27, -81, . . .
Solution: These are powers of three with alternating signs, so
the general term might be  1 3n1 .
n

Sums and Series
o Series: Given the infinite sequence a1 , a2 , a3 , a4 ,
of the terms a1  a2  a3 
 an 
, an ,
, the sum
is called an infinite series.
A partial sum is the sum of the first n terms a1  a2  a3 
 an .
A partial sum is also called a finite series or nth partial sum,
and is denoted S n .

Sigma Notation: The Greek letter  (sigma) can be used to denote a
sum when the general term of a sequence is a formula.
o Example: The sum of the first four terms of the sequence 3, 5,
7, 9, . . ., 2k  1 , . . . can be named
4
  2k  1.
1

Recursive Definitions: A sequence may be defined recursively or by
using a recursion formula. Such a definition lists the first term, or
the first few terms, and then describes how to determine the
remaining terms from the given terms.
o Example: Find the first 5 terms of the sequence defined by
a1  5, an1  2an  3, for n  1.
Solution:
a1  5
a2  a11  2a1  3  2  5   3  7,
a3  a21  2a2  3  2  7   3  11,
a4  a31  2a3  3  2 11  3  19,
a5  a41  2a4  3  2 19   3  35.
7.2
ARITHMETIC SEQUENCES AND SERIES

Arithmetic Sequences: A sequence is arithmetic if there exists a
number d, called the common difference, such that an 1  an  d for
any integer n  1.
o nth Term of an Arithmetic Sequence: The nth term of an
arithmetic sequence is given by an  a1   n  1 d , for any integer
n  1.

Example: Find the 14th term of the arithmetic sequence
4, 7, 10, 13, . . .
Solution:
a1  4, d  7  4  3, and n  14.
an  a1   n  1 d
a14  4  14  1  3  43.

Example: Which term is 301 from the sequence above?
Solution:
an  a1   n  1 d
301  4   n  1  3
301  4  3n  3
301  3n  1
300  3n
100  n.
The term 301 is the 100th term of the sequence.

Sum of the First n Terms of an Arithmetic Sequence
o Consider the arithmetic sequence 3, 5, 7, 9, . . . When we add
the first four terms of the sequence, we get S 4 , which is 3 + 5
+ 7 + 9, or 24. This sum is called an arithmetic series. To find
a formula for the sum of the first n terms, S n , of an arithmetic
sequence, we first denote an arithmetic sequence, as follows:
a1 ,  a1  d  ,  a1  2d  ,
,
 an  2d  ,  an  d  , an .
This term is 2 terms
back from the last
Sn  a1   a1  d    a1  2d  
This is the next
to last term.
  an  2d    an  d   an .
reversing the order gives us
Sn  an   an  d    an  2d  
  a1  2d    a1  d   a1.
adding these two sums we have,
2Sn   a1  an    a1  d    an  d    a1  2d    an  2d 

  an  2d    a1  2d    an  d    a1  d    an  a1 .
Notice that all of the brackets simplify to a1  an and
that a1  an is added n times. This gives us
2Sn  n  a1  an  or S n 
n
 a1  an 
2
So the sum of the first n terms of an arithmetic sequence is given by
Sn 
n
 a1  an 
2
7.3
GEOMETRIC SEQUENCES AND SERIES

GEOMETRIC SEQUENCE: A sequence is geometric if there is a
number r, called the common ratio, such that
an 1
 r,
an

an 1  an r ,
or
nth TERM OF A GEOMETRIC SEQUENCE: The nth term of a
geometric is given by
an  a1r n 1 ,

for any integer n  1.
SUM OF THE FIRST n TERMS: The sum of the first n terms of a
geometric sequence is given by
Sn 

for any integer n  1.
a1 1  r n 
1 r
,
for any r  1.
INFINITE GEOMETRIC SERIES: The sum of the terms of an
infinite geometric sequence is an infinite geometric series. For some
geometric sequences, S n gets close to a specific number as n gets very
large. For example, consider the infinite series
1 1 1 1
   
2 4 8 16

1

2n
LIMIT OR SUM OF AN INFINITE GEOMETRIC SERIES
o When r  1, the limit or sum of an infinite geometric series is
given by
S 
7.4

MATHEMATICAL INDUCTION
skip
a1
.
1 r
7.5
COMBINATORICS: PERMUTATIONS
7.6
COMBINATORICS: COMBINATIONS
7.7
THE BINOMIAL THEOREM
7.8
PROBABILITY