Download 7.1 - San Juan College

SECTION 7.1
SEQUENCES
SEQUENCES
A number sequence is an
arrangement of numbers in which
there is first number, a second
number, a third number, . . .
Example:
2,4,6,8,...
We call this an infinite sequence.
NOTATION:
In general, a number sequence is
denoted as:
a 1, a 2, a 3, a 4, a 5, . . .
The subscript denotes the
position of the term in the
sequence.
NOTATION:
A general term in the sequence is
denoted as:
an
An entire general sequence might
be denoted as:
{ a n}
EVALUATING TERMS IN A
SEQUENCE:
A sequence { pn } is determined by
the formula pn = n 2 - n + 1.
Evaluate p 3 and p 10
p3=32-3+1=7
p 10 = 10 2 - 10 + 1 = 91
WRITING THE FIRST SEVERAL
TERMS OF A SEQUENCE
Write down the first six terms of
the sequence:
a n 
 n  1
 

 n 
a1 = 0
a3 = 2/3
a5 =4/5
a2 = 1/2
a4 = 3/4
a6 = 5/6
USING THE GRAPHING
CALCULATOR
Find the first six terms of the
 n  1
sequence:
a  
n

 n


2nd List Ops
Choose Seq
seq(expression,variable,begin,end,
increment)
WRITING THE FIRST SEVERAL
TERMS OF A SEQUENCE
Write down the first six terms of the
sequence:

n - 1  2 
b  
n
(- 1)   
 n 

b1 = 2
b3 = 2/3
b5 = 2/5
b2 = - 1
b4 = -1/2
b6 = -1/3
FINDING AN EXPLICIT
FORMULA FOR NTH TERM
a 1 = 2 = 2(1)
a 2 = 4 = 2(2)
a 3 = 6 = 2(3)
a 4 = 8 = 2(4)
a n = 2(n)
Thus, if we need
to know the value
of the 39th term:
a 39 = 2(39) = 78
FINDING AN EXPLICIT
FORMULA FOR NTH TERM
a 1 = 0 = (1-1)2
a 3 = 4 = (3-1)2
Give an explicit
formula for the
sequence { a n } with
initial terms:
a 4 = 9 = (4-1)2
0, 1, 4, 9, 16, 25, . . .
a 2 = 1 = (2-1)2
a 5 = 16 = (5-1)2
a n = (n - 1)2
Use the explicit formula for { a n } to
determine the value of a 13
a n = (n - 1)2
a 13 = (13 - 1)2
a 13 = 144
THE FACTORIAL SYMBOL
If n > 0 is an integer, the factorial
symbol n! is defined as follows:
0! = 1
1! = 1
n! = n(n - 1) · . . . · 3 · 2 · 1
n! = n(n - 1)!
if n > 2
SEQUENCE FORMULAS
Seeing Patterns:
1 2 3 4 5 6 7
a) , , , , , , , . . .
3 4 5 6 7 8 9
b) 1, 3, 6, 10, 15, 21, 28, . . .
n
an =
n + 2
b1 = 1
b n = bn - 1 + n
Recursion Formula
EXAMPLE
Write down the first five terms of the
sequence defined by:
s1 = 1
sn = 4s n - 1
s2 = 4 · 1 = 4
s4 = 4 · 16 = 64
s3 = 4 · 4 = 16
s5 = 4 · 64 = 256
EXAMPLE:
Find the pattern in 1, 1, 2, 3, 5, 8, 13, . . .
f1 = f2 = 1 and fn = fn - 2 + fn - 1
This is called the Fibonacci Sequence
EXAMPLE:
Determine the first six terms of the
sequence:
an =
1   1 + 5 


2 
5  
n
1 - 5  
 
- 
 2  
n
EXAMPLE:
Determine the first six terms of the
sequence:
(bn - 2 + bn - 1 )
b1 = 0, b2 = 1, and bn =
,n > 2
2
b1 = 0
1
b3 =
2
5
b5 =
8
b2 = 1
3
b4 =
4
11
b6 =
16
Sums are usually denoted in sigma
notation:
n
a
k
= am + a m + 1 + am + 2 + . . . + a n
k
= a3+a4+a5 +a6
k=m
6
a
k=3
100

2 + 3 2 + . . . + 100 2
1
+
2
k =
k=1
2
EXAMPLE
100
1
=

k = 1k
1
1
1
1
1 +
+
+
+ ... +
2
3
4
100
We have no formula for this but, with
patience, could find the sum with a
calculator.
CONCLUSION OF SECTION 7.1