Honors Precalculus Notes--Chapter 10, Sequences, Induction, and Probability

, 10.1 Sequences and Summation Notation

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 the are
.

, ,
sequence


represented by

A ,
,
A 2
,
A3
,
Ay ,
. . .


, Ang . . .




Sequences whose domains consist
only of the first n positive integers are called finite sequences .




Example

write first four terms of the nth term term
the
sequence whose ,
or
general ,
is
given

an
=
2n -15


To find the first four terms ,
we replace n in the formula with 1,2 , 3, and 4 .




A
,
= 211 ) -15 Az
-
-
2. (2) + S a 3=213 ) -15 a 4=2143+5

=
2+5 = 4+5 = 6+5 =
8+5

= 7 = 9 = 11 =
13




Example

write first four terms of the nth term term
the
n
sequence whose ,
or
general ,
is
given
C- 1)

an
'
271

To find the first four terms ,
we replace n in the formula with 1,2 , 3, and 4 .




12 13 14
'
-
I - -
-




- - - -




A, =
2 't l Az :
22+1 A. 3=23+1 Ay
-

-
24+1
-
l l - l l
- - - -


=
2+1 =
4+1 =
8+1 =
16+1
l

f
l l
-
- - -
-




=
3 =
5 = =
17




Recursion Formulas

A recursion formula defines the nth term of a
sequence as a function of the previous term .




* Uses the previous term to find the next term




Example

Find the first four terms of the in which a. =3 an an =
2am -15 for n
? 2
sequence , .
.




=3 Az Zaz +5 Zaz +5
Zay -15
-




A Az Ay
-
- : -



, , , ,
. .
.




2A , -15 Zaz -15 Zaz
'
=




-15=213
= -




) -15 =
2111 ) -15 =
2127 ) -15

= 11 =
27 =
59