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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.
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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.
1, 4, 9, 16, 25,… is the sequence of square numbers.
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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.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, p, e, -110, …is a sequence without an obvious pattern. <= /span>
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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>
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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;
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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>
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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,
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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.
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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,
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>
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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>
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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.
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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.
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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
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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.)
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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
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.)
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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.
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.)
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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.
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.)
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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.
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.)
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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 =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.)
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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 =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.)
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Example C:<= /span>
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=