Avon High School
Section: 11.1
ACE COLLEGE ALGEBRA II - NOTES
Sequences and Summation Notation
Mr. Record: Room ALC-129
Semester 2 - Day 36
Sequences
In the thirteenth centry, an Italian mathematician named Leonardo of Pisa – also
known as Fibonacci, investigated a rather interesting string of numbers. These
numbers, separated by commas, are referred to as a sequence. Fibonacci’s sequence
went like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 87, …
What pattern do you see with this sequence?
So many things in nature feature these numbers from fruits to flowers. The ratio of
two consecutive Fibonacci numbers can be seen in the art and archtictecture of the
Renaissance period.
It’s sometimes helpful to think of a sequence as a function. For the Fibonacci
sequence, we would see
Therefore, f (1)  1, f (2)  1, f (3)  2, f (4)  3, f (5)  5...
We can also write the terms as such: a1  1, a2  1, a3  2, a4  3, a5  5,...
an represents the nth term or general term of the sequence. The entire sequence is denoted by an  .
Definition of a Sequence
An infinite sequence an  is a function whose domain is the set of positive integers. The function
values, or terms, of this sequence are represented by:
a1 , a2 , a3 , a4 , an ,
Sequences whose domains consist only of the first n positive integers are called finite sequences.
Example 1
Writing Terms of a Sequence from the General Term
Write the first four terms of the sequence whose nth term, or general term, is given.
(1) n
a. an  2n  5
b. an  n
2 1
Note the difference between the graph of the function, f ( x ) 
1
1 
and the sequence an     .
x
n
Recursion Formulas
To put it simply, a recursion formula defines the nth term of a sequence as a function of its previous term.
Example 2 illustrates this well.
Using a Recursion Formula
Example 2
Find the first four terms of the sequence in which a1  3 and an  2an1  5 for n  2.
Factorial Notation
Factorial Notation
If n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n
down through 1.
n !  n(n  1)(n  2) (3)(2)(1)
0! (zero factorial), by definition, is 1.
0! = 1
Activity: Find the values of the first six positive integers.
Example 3
1! =
2! =
3! =
4! =
5! =
6! =
Finding Terms of a Sequence Involving Factorials
Write the first four terms of the sequence whose nth term is an 
20
.
(n  1)!
Evaluating Fractions with Factorials
Example 4
a.
Evaluate each factorial expression.
( n  1)!
b.
n!
14!
2!12!
c.
n!
(n  1)!
Summation Notation
Summation Notation
The sum of the first n terms of a sequence is represented by the summation notation
n
a
i 1
i
 a1  a2  a3  a4 
an .
where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.
Using Summation Notation
Example 5
Expand and evaluate the sum.
6
a.
5
 2i 2
b.
i 1
Example 6
5
 (2k  3)
c.
k 3
i1
Writing Sums in Summation Notation
Express each sum using summation notation.
a. 12  22  32 
 92
1 1 1
b. 1    
2 4 8
4

1
2n1
Properties of Sums
Let n be any integer such that n > 1 and c be any real number.
1.
2.
3.
n
n
i 1
i 1
 cai  c ai
n
n
n
i 1
i 1
i 1
n
n
n
i 1
i 1
i 1
  ai  bi   ai   bi
  ai  bi   ai   bi