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Click to edit Master title style
Clic= k to edit Master text styles
Sec= ond level
Thi= rd level
Fou= rth level
Fif= th level
‹date/time›&= #13;
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------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/master06.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADxwAAAAYAHAAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwCeQChAAMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAOAAAAJAMFAP///////3gA oAB4AKAA////////CAAAAPoCAAAAAAAAAAAAAAQAAAAtAQIABwAAAPwCAAD///8AAAAEAAAALQED AAQAAAAnAf//HAAAAPsC+f8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAEAAAALQEEAAQAAAAuARgABAAAAAIBAQAFAAAACQIAAAACFQAAADIKBwA6AAkAAABT ZXF1ZW5jZXMABQAEAAQABAAEAAQABAAEAAMABAAAAC4BAAAcAAAA+wIQAAcAAAAAALwCAAAAAAEC AiJTeXN0ZW0AAAAAAAAAAAAAGAAAAAIAAADAxLQH5AQAAAQAAAAtAQUABAAAAPABBAADAAAAAAA= ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0077.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Sequences= <= /div>
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A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Sequences= <= /div>
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A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0082.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Def= inition: A sequence is the list of outputs f(n) of a function f where n =3D 1, 2, 3,= 4, ….. We write f(n) as fn.
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0083.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Def= inition: A sequence is the list of outputs f(n) of a function f where n =3D 1, 2, 3,= 4, ….. We write f(n) as fn.
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
A sequence may be listed = as f1, f2 , f3 , <= /span>= 230;
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0084.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Def= inition: A sequence is the list of outputs f(n) of a function f where n =3D 1, 2, 3,= 4, ….. We write f(n) as fn.
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
A sequence may be listed as f1, f2 , f3 , <= /span>= 230;
f100 =3D= 100th number on the list, <= /span>
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0085.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Def= inition: A sequence is the list of outputs f(n) of a function f where n =3D 1, 2, 3,= 4, ….. We write f(n) as fn.
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
A sequence may be listed as f1, f2 , f3 , <= /span>= 230;
f100 =3D= 100th number on the list, &= #13;
fn =3D the n’th number on the l= ist,
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0086.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
 
Def= inition: A sequence is the list of outputs f(n) of a function f where n =3D 1, 2, 3,= 4, ….. We write f(n) as fn.
Sequences= <= /div>
Example A:
1, 3, 5, 7, 9,= 230; is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
A sequence may be listed as f1, f2 , f3 , <= /span>= 230;
f100 =3D= 100th number on the list, &= #13;
fn =3D the n’th number on the list,
fn-1 =3D= the (n – 1)’th number on the list or the number before fn.
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0087.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
Sequences= <= /div>
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Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
Sequences= <= /div>
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Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
b. For the sequen= ce of even numbers 2, 4, 6, 8, …       
    f3=3D 6,  = f4 = =3D 8,  f5 = =3D 10,
Sequences= <= /div>
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0089_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhcAArAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABt ACUAgAAAAAAAAALQhI+hge0Po5y02rby3bz7vRzhR5ZmpWDMybZnmqzuTKOyWOf6Ayf7v+sBh4CR S0gMao63ZC6ENEWdTFHTx5lSXzdtUbaEeLekpheWOV/JJaT6a42Mqek1TyVJW+3GbV3vMdcHxzNX 8zfYYTiolTgEyLJ41YhoyJZnh6fCUFl5+WEZ1en4aakJSlnqeYGoyJf5uGrxR2P6ORF2aHsrBhu5 y3sJHEw2TEzne3y7uGesDCLISfusE034lUyNisvorI1JkUj6/UOZTV5VeI7+y85SAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
b. For the sequen= ce of even numbers 2, 4, 6, 8, …       
    f3=3D 6,  = f4 = =3D 8,  f5 = =3D 10,
    and a formula for fn is f =3D 2*n 
Sequences= <= /div>
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Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
b. For the sequen= ce of even numbers 2, 4, 6, 8, …       
    f3=3D 6,  = f4 = =3D 8,  f5 = =3D 10,
    and a formula for fn is f =3D 2*n 
Sequences= <= /div>
c. For the sequen= ce of odd numbers 1, 3, 5, …    a general formula is = fn=3D 2n – 1.
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0091_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAAdAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ ABwAgQAAAAAAALvg4wECAwJWjI+py+0topy02otzNbr7r3HgSHZiiabRqbYg68YZLNcUbed4Xu98 7Pu1gsIUsVg6IkfK5afpNAWiMigVY71astrbtGv8gpPiMbNsfqLTUvb5AY/L4QUAOw== ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0092.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
b. For the sequen= ce of even numbers 2, 4, 6, 8, …       
    f3=3D 6,  = f4 = =3D 8,  f5 = =3D 10,
    and a formula for fn is f =3D 2*n 
Sequences= <= /div>
c. For the sequen= ce of odd numbers 1, 3, 5, …    a general formula is = fn=3D 2n – 1.
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Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
b. For the sequen= ce of even numbers 2, 4, 6, 8, …       
    f3=3D 6,  = f4 = =3D 8,  f5 = =3D 10,
    and a formula for fn is f =3D 2*n 
Sequences= <= /div>
c. For the sequen= ce of odd numbers 1, 3, 5, …    a general formula is = fn=3D 2n – 1.
d. For the sequen= ce of odd numbers with alternating &nbs= p;
    signs -1, 3, -5, 7, -= 9, …fn =3D (-1)n(2n – 1)
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Example B: <= /span>
a. For the sequen= ce of square numbers 1, 4, 9, 16, …       
    f3 =3D 9,  = f4=3D 16,  f5 =3D 25,
    and a formula for fn = is fn =3D n.
b. For the sequen= ce of even numbers 2, 4, 6, 8, …       
    f3=3D 6,  = f4 = =3D 8,  f5 = =3D 10,
    and a formula for fn is f =3D 2*n 
Sequences= <= /div>
c. For the sequen= ce of odd numbers 1, 3, 5, …    a general formula is = fn=3D 2n – 1.
d. For the sequen= ce of odd numbers with alternating &nbs= p;
    signs -1, 3, -5, 7, -= 9, …fn =3D (-1)n(2n – 1)
A sequence whose signs alternate is called an = alternating sequence as= in part d.
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Summation Notation
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In mathematics, the Greek letter “S” (sigma) means
“to add”. 
Summation Notation
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In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
Summation Notation
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In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s. 
Summation Notation
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In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
Summation Notation
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0100.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Given a list, let's say, = of 100 numbers, f1, f2, f= 3,.., f100,
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0101.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk
k =3D 1
100
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
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Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk 
k =3D 1
100
A variable which is called the
“index” variable, in this case k.
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0102_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAAS ABsAgAAAAAAAAAIxhI9poX3BXIJRLlotzppahH0XJ3nidY6peUIpwIruOsu1/cbfvfOl/8sBG7qO kCQpAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0103.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk  
k =3D 1
100
A variable which is called the
“index” variable, in this case k.
k begins with the bottom number
and counts up (runs) to the top number.
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0103_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAAS ABsAgAAAAAAAAAIxhI9poX3BXIJRLlotzppahH0XJ3nidY6peUIpwIruOsu1/cbfvfOl/8sBG7qO kCQpAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0104.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk 
k =3D 1
100
A variable which is called the
“index” variable, in this case k.
k begins with the bottom number
and counts up (runs) to the top number.
The beginning number
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0104_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAAS ABsAgAAAAAAAAAIxhI9poX3BXIJRLlotzppahH0XJ3nidY6peUIpwIruOsu1/cbfvfOl/8sBG7qO kCQpAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0104_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAUABgA6 AB8AgAAAAAAAAAJVjA+py+0P3zGx2isn3jzP0IUdCHziyZnoWqns27jwXGo0bd9zrrNyj/oBRcLh iGcMFZMYJDOFeCqdUsuyCrliHdoto+tVUMPcKDkCJqfDa297+8ZSCgA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0105.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk 
k =3D 1
100
A variable which is called the
“index” variable, in this case k.
k begins with the bottom number
and counts up (runs) to the top number.
The beginning number
The ending number<= /div>
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0105_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAAS ABsAgAAAAAAAAAIxhI9poX3BXIJRLlotzppahH0XJ3nidY6peUIpwIruOsu1/cbfvfOl/8sBG7qO kCQpAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0105_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRwBhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAQBC AF4AgAAAAAAAAAKshI+pyxcfmpy0Jgit3hVHDoaIJ5YgaaaU96nuwr7yhc22gd5yrqtx7/oBU7xh qWi0tADIZOfTdK5YS+kman1msiEq9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8PGCg4SFho eIiYqLgn9Ial9ojmBQdVMxeZhilpKad5xtm5FSoa52lmWoZKpjoGWkpK6eoo68YqZhuGC0ZbCzvr 2wvcpvtFzGWc9QBQAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0106.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk =3D f1    = f2    = f3  …       f99    f100
k =3D 1
100
A variable which is called the
“index” variable, in this case k.
k begins with the bottom number
and counts up (runs) to the top number.
The beginning number
The ending number<= /div>
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0106_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAAS ABsAgAAAAAAAAAIxhI9poX3BXIJRLlotzppahH0XJ3nidY6peUIpwIruOsu1/cbfvfOl/8sBG7qO kCQpAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0106_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRwBhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAQBC AF4AgAAAAAAAAAKshI+pyxcfmpy0Jgit3hVHDoaIJ5YgaaaU96nuwr7yhc22gd5yrqtx7/oBU7xh qWi0tADIZOfTdK5YS+kman1msiEq9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8PGCg4SFho eIiYqLgn9Ial9ojmBQdVMxeZhilpKad5xtm5FSoa52lmWoZKpjoGWkpK6eoo68YqZhuGC0ZbCzvr 2wvcpvtFzGWc9QBQAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0107.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
Summation Notation
Giv= en a list, let's say, of 100 numbers, f1, f2, f= 3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may= be written in the S- notation as:
S fk =3D f1+ f2+ f3+ … + f99+ f= 100
k =3D 1
100
A variable which is called the
“index” variable, in this case k.
k begins with the bottom number
and counts up (runs) to the top number.
The beginning number
The ending number<= /div>
In mathematics, the Greek letter “S” (sigma) means
“to add”.  Hence, “Sx” means to add the x’s,
“S(x*y)” means to add the x*y’s.  (Of course= the x's and xy's have to be given in the context.)
------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0107_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAAS ABsAgAAAAAAAAAIxhI9poX3BXIJRLlotzppahH0XJ3nidY6peUIpwIruOsu1/cbfvfOl/8sBG7qO kCQpAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0107_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRwBhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAQBC AF4AgAAAAAAAAAKshI+pyxcfmpy0Jgit3hVHDoaIJ5YgaaaU96nuwr7yhc22gd5yrqtx7/oBU7xh qWi0tADIZOfTdK5YS+kman1msiEq9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8PGCg4SFho eIiYqLgn9Ial9ojmBQdVMxeZhilpKad5xtm5FSoa52lmWoZKpjoGWkpK6eoo68YqZhuGC0ZbCzvr 2wvcpvtFzGWc9QBQAAA7 ------=_NextPart_01C8FFDF.399783C0 Content-Location: file:///C:/30889913/1Sequences_files/slide0108.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sequences
S f=3D<= /div>
k=3D4=
8
S a=3D =
i=3D2=
5
S xyj =3D
j=3D6=
9
S a=3D =
j=3Dn=
n+3
Summation Notation
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Example C:<= /span>
S f=3D f4+f5+f6+f7+ f
k=3D4=
8
S a=3D =
i=3D2=
5
S xyj =3D
j=3D6=
9
S a=3D =
j=3Dn=
n+3
Summation Notation
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Example C:<= /span>
S f=3D f4+f5+f6+f7+ f
k=3D4=
8
S a=3D a2+a3+a4+a5
i=3D2=
5
S xyj =3D
j=3D6=
9
S a=3D =
j=3Dn=
n+3
Summation Notation
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Example C:<= /span>
S f=3D f4+f5+f6+f7+ f
k=3D4=
8
S a=3D a2+a3+a4+a5
i=3D2=
5
S xyj =3D xy6+x7y+ xy8+x9y
j=3D6=
9
S a=3D =
j=3Dn=
n+3
Summation Notation
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Example C:<= /span>
S f=3D f4+f5+f6+f7+ f
k=3D4=
8
S a=3D a2+a3+a4+a5
i=3D2=
5
S xyj =3D xy6+x7y+ xy8+x9y
j=3D6=
9
S a=3D an+an+1+an+2+an+3
j=3Dn=
n+3
Summation Notation
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Example C:<= /span>
S f=3D f4+f5+f6+f7+ f
k=3D4=