Lecture 6 ethics for information technology
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Sequences, Series, and the Binomial Formula
Lecture 6
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Section 6.1Sequences
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SequenceA sequence is a set of numbers arranged
in some order.Each number is labeled with a variable, such as
a.The variable is indexed with a natural number that
tells its position in the sequence.
The numbers a1, a2, a3,. . . are the terms of the sequence.The first term in the sequence is a1, the second term
a2, the third term a3, and so on.
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General Term of a Sequence
Many sequences follow some sort of pattern.The pattern is usually described by the nth term of
the sequence.This term, an, is called the general term of the
sequence.
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Finite Sequence
A finite sequence has a specific number of terms and so it has a last term.
An infinite sequence does not have a last term.
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Sequence Notation
The notation {an} is often used to represent a
sequence whose nth term is an.
The { } indicate that it is a sequence.
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Example – sequence notationFind the first five terms of the sequence
7
2 3n
21 1 3 1 3 2a
22 2 3 4 3 1a
23 3 3 9 3 6a
24 4 3 16 3 13a
25 5 3 25 3 22a
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Recursion Formula
A recursion formula defines a sequence in terms of one or more previous terms.A sequence that is specified by giving the first
term, or the first few terms, and a recursion formula is said to be defined recursively.
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Section 6.2 Arithmetic and Geometric Sequences
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Arithmetic SequenceAn arithmetic sequence, or arithmetic
progression, is a sequence where each term is obtained from the preceding term by adding a fixed number called the common difference.If the common difference is d, then an
arithmetic sequence follows the recursion formula an = an-1 + d.
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Terms of an Arithmetic Sequence
If a1 is the first term of an arithmetic sequence, an the nth term and d the common difference,
then an = a1 + (n - 1)d.
If an-1 and an are consecutive terms of an arithmetic sequence, then d = an - an-1.
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Geometric SequenceA geometric sequence, or geometric
progression, is a sequence where each term is obtained by multiplying the preceding term by a fixed number called the common ratio.If the common ratio is r, then a geometric
sequence follows the recursion formula: an = ran-1.
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Terms of a Geometric SequenceIf a1 is the first term of a geometric
sequence, an the nth term, and r the common ratio, then an = rn-1a1
If an-1 and an are consecutive terms of a geometric sequence, then
1
n
n
a
ar
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Example – Terms of a SequenceIf the first term is 5, the last (nth) term is
-139, and d = -6, how many terms are there?
14
1 ( 1)na a n d
There are 25 terms.
139 5 ( 1)( 6)n 144 6 6n
150 6 ; 25n n
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Section 6.3 Series
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Series
The sum of the terms of a sequence is called a series.The series a1 + a2 + a3 + a4 + a5 is a finite
series with five terms.
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Infinite Series
A series of the form a1 + a2 + a3 + a4 + · · ·is an infinite series.An infinite series has an infinite or endless
number of terms.
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Summation NotationSummation or sigma notation means
sum. a indicates Here -
n
n
kk aaaaa 321
1
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Limits of Summation
The letter k in is called an index of summation.
The summation begins with k = 1 as is indicated below the and ends with k = n as indicated above the .
The numbers below and above theare the limits of summation.
n
kka
1
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Property 1 of Summation Notation
If a is a constant then
n
k
n
k
kaak11
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Property 2 of Summation Notation
If a is a constant then
naan
k
1
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Property 3 of Summation NotationIf x and y are constants then
n
k
n
k
n
k
yxyx1 11
)(
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Partial Sum
If Sn = a1 + a2 + a3 + · · · + an =then the number Sn is called the nth partial sum of the series.
The sequence S1, S2, S3, . . . , Sn is called the sequence of partial sums.
n
kka
1
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Arithmetic Series
An arithmetic series is a series formed by the partial sums of an arithmetic sequence.
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Sum of First n Terms of an Arithmetic SequenceThe sum Sn of the first n terms of an
arithmetic sequence is
where a1 is the first term and an is the nth term.
2
)( 1 nn
aanS
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Geometric Series
A geometric series is a series formed by the partial sums of a geometric sequence.
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Sum of First n Terms of a Geometric Sequence
The sum Sn of the first n terms of a geometric sequence is
where a1 is the first term and r is the common ratio.
r
raS
n
n
1
11
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Compound Interest
A = P(1 + i)n
where A is the amount after n interest periods, P is the principal or initial amount invested, and i is the interest rate per interest period expressed as a decimal.
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Example-Summation Notation
Evaluate
29
3
0
2
3 5
k
k k
3
0
2
3 5
k
k k
0 1 2 32 2 2 2
3 0 5 3 1 5 3 2 5 3 3 5
1 2 4 8
0 5 3 5 6 5 9 5
11 4 2
5
445
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Section 6.4 Infinite Geometric Series
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Infinite Series, 1
A series that does not have a last term is called an infinite series.
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Sequence of Partial Sums
series. infinite the of sum the is that
say then we, call will wethat number some
gapproachin be to seems larger, and larger
gets as that, happens it If
is The
series the of the called is
S
S
S
nSSSS
a
nS
n
n
k k
n
.,,,,
.
321
1
sums partial of sequence
sum partial th
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Convergent Series
If the partial sums of an infinite series approach a finite limit, we say that the series converges or is a convergent series.
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Divergent Series
A series that does not converge is said to diverge or to be a divergent series.All arithmetic series diverge.
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Example – Infinite SeriesFind the fraction that has the repeating decimal form 0.232323 . . . .:
35
0
0.23(0.01)n
n
This decimal can be thought of as the series 0.23 + 0.0023 + 0.000023 +…, which is the geometric series
In this series, so 1 0.23, 0.01,a r 1
1
as
r
0.23 0.23 23
1 0.01 0.99 99
Thus, decimal 0.232323 . . is equivalent to the fraction23
99
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Sum of an Infinite Geometric Series
diverges. series the thenIf
sum thehas
seriesgeometric infinite the thenIf
,1
.1
,1
1
13
12
111
r
r
aS
rarararaa
r
n
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Section 6.5 The Binomial Theorem
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Expansions of x + ySome expansions of x + y are
(x + y)0 = 1(x + y)1 = x + y(x + y)2 = x2 + 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5
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Pascal’s Triangle
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Binomial Formula, 1
nn
nnnn
yyxnnn
yxnn
ynxxyx
33
221
32
)2()1(2
)1()(
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n Factorial
n! = n(n - 1)(n - 2)(n - 3) · · · (3)(2)(1)
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r
n
)!(!
!
rnr
n
represents
r
n
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Example - Factorials
Determine 5! And 8!
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5! 5 4 3 2 1 120
8! 8 7 6 5 4 3 2 1 40320
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Binomial Formula, 2 (1 of 2)
rrnn
r
nn
n
nnnn
yxr
n
yxyn
n
yxn
yxn
yxn
xyx
0
1
33
221
1
3
21)(
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Binomial Formula, 2 (2 of 2)
!
)1()2()1(
!)!(
!
r
rnnnn
rrn
n
r
n
is term general thewhere
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Binomial Series
32
132
!3
)2()1(
!2
)1(11
.1
1321)1(
xnnn
xnn
nxx)(
nx
n
xxn
nx
nx
nnxx
n
nnn
series. infinite the
get younumber, rational a or negative isIf
if number realany for valid is series binomial The