Download Arithmetic Sequences

Sequences and the
Binomial Theorem
•Sequences
•Arithmetic Sequences
•Geometric Sequences & Series
•Binomial Theorem
Sequences
A sequence is a function whose
domain is the set of positive
integers.
n
n
Write down the first 3 terms of  1  n 
2 
1
1
1 1 
a1    1  1   1   
2 
2
2
2
2
2 1
a2    1  2   1  
2 
4 2
3
3
3
3
a3    1  3   1   
2 
8
8
An important common usage is in binomial coefficient.
The binomial coefficient has, unfortunately, three common notations:
 n
n!
C (n, r )  n C r    
.
r!(n  r )!
r
Evaluate: 6!
6!  6  5  4  3  2  1
 720
Sequences in which the first (or first
few) terms are assigned values and
the nth term is defined by a formula
that involves one (or more) terms
preceding it is a recursively defined
sequence.
a1  100
a2  1.05a1  1.05(100)  105
a3  1.05a2  1.05(105)  110.25
a4  1.05a3  1.05(110.25)  115.7625
Summation Notation
n
 ak  a1  a2  an
k 1
Find the sum of the sequence
3


 2k  4k  1
3
k 1

2

3
3
3
k 1
3
k 1
 2k  4k  1   2k   4k  1
2
k 1
k 1
3
2
2
3
 2  k  4  k  1


k 1
k 1
k 1
 2 1  2  3  41  2  3  1  1  1
2
2
2
 214   4 6  3  7
Arithmetic Sequences
An arithmetic sequence is defined
as
a1  a
a n 1  a n  d
First term: a1  2 (1)  7  5
an1  2(n  1)  7  2n  2  7  2n  5
d  an 1  an  2n  5  ( 2n  7)
 2n  5  2n  7  2
Common difference does not depend on
n, therefore the sequence is arithmetic.
Theorem: nth Term of an Arithmetic
Sequence
For an arithmetic sequence an  whose first
term is a and whose common difference is d ,
the nth term is defined by the formula
an  a  n 1d
a1  4
d  a2  a1  1  4  3
an  a  n  1d  4  n  1 3
a20  4  20  1  3
 53
The 6th term of an arithmetic sequence is 31.
The 19th term is 109. Find the first term and
the common difference. Give a recursive
formula for the sequence.
an  a  ( n  1) d
a6  31  a  5d
a19  109  a  18d
 78  13d
d 6
an  a  ( n  1) d
a6  31
a  5d  31
d 6
a  5( 6)  31
a 1
an  1  ( n  1) 6
 1  6n  6  6n  5
d  an 1  an
an 1  an  d
a n 1  a n  6
Theorem: Sum of n Terms of an
Arithmetic Sequence
Let an  be an arithmetic sequence with the
first term a and common difference d . The
sum Sn of the first n terms of an  is
n
n
Sn  2a  (n  1)d   a  an 
2
2
Find the sum of the first 30 terms of the
sequence {7n + 2}. That is, find
9 + 16 + 23 + . . . + 212
a  9 , a30  212 , n  30
30
S30  9  212
2
 3315
Geometric Sequences
&
Geometric Series
A sequence is geometric when the
ratio of successive terms is always the
same nonzero number. A geometric
sequence is defined recursively as
a1  a
an 1  ran
Determine if the following sequence is
geometric
3, 15, 75, 375, . . .
15
5
3
75
5
15
375
5
75
The sequence is geometric with a common
ratio of 5.
Theorem: nth Term of a Geometric
Sequence
For a geometric sequence an  whose
first term a and whose common ratio
is r , the nth term is determined
by the formula
an  ar
n 1
, r0
First term: 4
an  ar
n 1
1

a11  4 
 2
1
Common ratio:
2
1

 4 
 2
111
n 1
1

 4 
 2
10
1

256
Theorem: Sum of First n Terms of a
Geometric Sequence
Let an  be a geometric sequence with first term
a and common ratio r. The sum Sn of the first n
terms of an  is
1 r
Sn  a
, r  0, 1
1 r
n
Find the sum of the first 10 term s
  1 n 
of the sequence    . That is, find
  5 
1 1
1
1



   
 5
5 25 125
10
Geometric with a  1 , r  1
5
5
 
10
 
1
1
1

1

1
1
5
5
S10  


4
5 1 1
5
5
5
10
 0.249999974
Theorem: Amount of an Annuity
If P represents the deposit made in dollars at
each payment period for an annuity at i
percent interest per payment period, the
amount A of the annuity after n payment
periods is
1  i  1

A P
n
i
Suppose Yola deposits $500 into a Roth IRA
every quarter (3 months). What will be the
value of the account in 25 years assuming it
earns 9% per annum compounded quarterly?
0.09
i
 0.0225, n  25( 4)  100, P  $500
4
1  i  1
1  0.0225


A P
 $500
n
i
100
0.0225
 $183,423.25
1
An infinite sum of the form
n1
a  ar  ar ar 
with first term a and common ratio r , is called
an infinite geometric series and is denoted by
2

 ar
k 1
k 1
Theorem: Sum of an Infinite Geometric
Series
If r  1, the sum of the infinite geometric

series  ar
k 1
k 1
is

 ar
k 1
k 1
a

1 r
6 12
Find the sum of 3   
5 25
a  3,
r 1
6
6 2
5
r


3 15 5
6 12
3    
5 25
3
2
1
5
5
Mathematical Induction
Theorem: Principle of Mathematical Induction
Suppose the following two conditions are satisfied
with regard to a statement about natural numbers:
CONDITION I: The statement is true for the
natural number 1.
CONDITION II: If the statement is true for some
natural number k, it is also true for the next natural
number k + 1.
Then the statement is true for all natural numbers.
CONDITION I: Show true for n = 1
1(1  1) k 11(2)
1
k  1  22
2
2
CONDITION II: Assume true for some
number k, determine whether true for k + 1.
k ( k  1)
Assume: 1  2  3 k 
2
( k  1)( k  2)
Show: 1  2  3 k  ( k  1) 
2
( k  1)( k  2)
1

2

3



k

(
k

1
)




2
k ( k 1)
2
k ( k  1) 2( k  1)
k ( k  1)

 ( k  1) 
2
2
2
k  3k  2 (k  1)(k  2)
k  k  2k  2



2
2
2
2
2
The Binomial Theorem
Definition: Binomial Coefficient Symbol
If j and n are integers with 0  j  n,
 n
the symbol   is defined as
 j
n!
 n
 
 j j !n  j !
 7
Evaluate:  
 3
7!
7!
 7

 
 3 3!7  3! 3!4!
7  6  5  4!

3  2  1  4!
 35
The Binomial Theorem
Let x and a be real numbers. For any positive
integer n, we have
n n n n1
n j n j
n n
x  a    x    ax   a x   a
0
1
 j
n
n
n n j j
   x a
j 0  j
n
 4
4  4
3
3y  5    3y    53y
 0
1 
4
 4 2
 4 4
2  4 3
   5 3 y     5 3 y     5
 2
 3
 4
 81y4  4  5 27 y3  6 25 9 y2  4 125 3y  625
 81y  540 y  1350 y  1500 y  625
4
3
2