SEQUENCES AND SERIES
Section 12-5
Sigma Notation and the nth term
SIGMA NOTATION OF A SERIES

For any sequence a1, a2, a3 … , th e su m of th e first
k term s m ay be w ritten
w h ich is read “th e
k

n 1
a
n
summation from n=1 to k of a . Thus,
k


n 1
a
n
= a1 + a2 + a3 + … + ak
where k is an integer value
EXAMPLE #1
Write each expression in expanded form and then

7
find the sum.
1n
b
2( )

(3

1)

1.
2. n 1 3


b 4
7
(3 1)= 82+244+730+2188=3244
b

b 4
1n
n 1 2(3 ) = 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + … .


This is an infinite geometric series r=1/3 a1=2/3
so S = 2/3 / 1-1/3 = 1
EXAMPLE #2


E xp ress th e series 16+ 19+ 22+ 25+ … + 61 u sin g
Sigma notation.
I notice that each term is three more than the
previous term. So I can write it as n+3. Since
the first term is 16, n starts at 13. 5 8

n3
n 1 3

Another way I can look at it is would be looking
for a pattern. Notice 3(5) + 1 = 16 3(6) + 1 = 19
3(7)+1=22
20
 3n  1
n 5
N FACTORIAL


The expression n! (n factorial) is defined as
follows for n, an integer greater than zero.
n! = n(n-1)(n-2)(n-3)… 1
EXAMPLE # 3

Express the series -4/1 + 16/2 - 64/6 + 256/24
in sigma notation.
Notice the signs alternate beginning with a
n
negative sign. This indicates (-1) . We also notice
that the numerator is powers of 4 and the
denominator is n!
So we have
n
4
n
(

1)
n1 n!
4
HW # 43
Section
12-5
Pp. 798-800
#14-35 all, 47a.,54