Section 8.1 – Sequences and Sigma Notation
is represented by an 
Infinite sequence: a1 , a2 ,
Finite sequence: a1 , a2 ,
, an is represented by an  where n is a positive integer
Sequences are represented by a formula for the nth term.
Example 1 -
an   2n  1
represents the sequence 3, 5, 7, … since a1 = 2(1) + 3, etc.
n

1 1 1
  1 

Example 2 - an   
   , , ,
2 4 6

 2n 

recursive definition –
the nth term is written in terms of previous terms, given a starting value
Example 3 - Given the sequence an  an1  2, a1  1 , the terms are found as follows:
a2  a1  2  1  2  3; a3  a2  2  3  2  5; etc. so the terms are 1, 3, 5, 7, …
On a TI-84+ graphing calculator,  (LIST) – OPS –with sequence MODE gives
seq(2n + 1, n, 1, 5) yielding {3 5 7 9 11}.
Factorial notation – n! = (n) (n – 1) … (2) (1). For example, 5! = (5) (4) (3) (2) (1) = 120.
Evaluating factorial expressions –
n  n  1 n  2 
n!

 n  1!  n  1 n  2 
or
 2 1  n
 2 1
n  n  1 !
n!

n
 n  1!  n  1!
Examples -
 20 19 18!  190,
20!

2!18!
 2 18!
 2n  1!   2n  1 2n !  2n  1
 2n  !
 2n  !
Sigma Notation –
n
a
i 1
i
 a1  a2 
 an where i is the index of summation, 1 is the lower limit of summation, and
n is the upper limit of summation.
Examples i
2
3
 1  1  1  1

        
3  3  3  3
i 2 
4
5

i 1
4
 i  2 !  1  2 !   2  2 !  3  2 !   4  2 !  5  2 !  110
i!
1!
2!
3!
4!
5!
On a TI-84+ graphing calculator,  (LIST) – MATH –with sequence MODE gives
sum(seq((n + 2)!/n!, n, 1, 5) yields 110.
Section 8.2 – Arithmetic Sequences
An arithmetic sequence is a sequence in which the difference between any two consecutive terms
is a fixed amount called the common difference d. For example, d  a2  a1; d  a3  a2
The formula for the nth term of an arithmetic sequence given the first term and common
difference is an  a1   n  1 d
Examples –
Find a2 given a1  300 and d  50
a2  300   2  1 50   350.
Find a16 given a1  9 and d  2
a16  9  16  1 2   39.
The formula for the sum of the first n terms of an arithmetic sequence given the first term, the
n
nth term, and n is Sn   a1  an 
2
Examples –
Find 2  4  6  8 
 200
n  100, a1  2, a100  200
100
S100 
 2  200   10100
2
sum (seq (2n, n, 1, 100)  10100
20
Find
  6i - 4 
i 1
n  20, a1  2, a20  116
20
 2  116   1180
2
sum (seq (6n - 4, n, 1, 20)  1180
S20 