2. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Sequences
3. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
4. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, , e2, -110, …is a sequence without an obvious
pattern.
6. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a
function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, , e2, -110, …is a sequence without an obvious
pattern.
7. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a
function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, , e2, -110, …is a sequence without an obvious
pattern.
A sequence may be listed as f1, f2 , f3 , …
8. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a
function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, , e2, -110, …is a sequence without an obvious
pattern.
A sequence may be listed as f1, f2 , f3 , …
f100 = 100th number on the list,
9. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a
function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, , e2, -110, …is a sequence without an obvious
pattern.
A sequence may be listed as f1, f2 , f3 , …
f100 = 100th number on the list,
fn = the n’th number on the list,
10. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a
function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:
1, 3, 5, 7, 9,… is the sequence of odd numbers.
1, 4, 9, 16, 25,… is the sequence of square numbers.
5, -2, , e2, -110, …is a sequence without an obvious
pattern.
A sequence may be listed as f1, f2 , f3 , …
f100 = 100th number on the list,
fn = the n’th number on the list,
fn-1 = the (n – 1)’th number on the list or the number
before fn.
11. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
Sequences
12. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
Sequences
13. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
Sequences
14. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2*n
Sequences
15. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, …
a general formula is fn= 2n – 1.
16. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, …
a general formula is fn= 2n – 1.
17. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, …
a general formula is fn= 2n – 1.
d. For the sequence of odd numbers with alternating
signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
18. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, …
a general formula is fn= 2n – 1.
d. For the sequence of odd numbers with alternating
signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
19. Example B:
a. For the sequence of square numbers 1, 4, 9, 16, …
f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, …
f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, …
a general formula is fn= 2n – 1.
d. For the sequence of odd numbers with alternating
signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
A sequence whose signs alternate is called an
alternating sequence as shown in part d.
21. In mathematics, the Greek letter “ ” (sigma) means
“to add”.
Summation Notation
22. In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’s,
Summation Notation
23. In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’s,
“ (x*y)” means to add the x*y’s.
Summation Notation
24. In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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
25. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
26. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fkk = 1
100
In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
27. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fkk = 1
100
A variable which is called the
“index” variable, in this case k.
In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
28. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fkk = 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 “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
29. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fkk = 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 “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
30. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fkk = 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
In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
31. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fk = f1 f2 f3 … f99 f100k = 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
In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
32. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
their sum f1 + f2 + f3 ... + f99 + f100 may be written in
the - notation as:
fk = f1+ f2+ f3+ … + f99+ f100k = 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
In mathematics, the Greek letter “ ” (sigma) means
“to add”. Hence, “ x” means to add the x’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.)
34. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai =
i=2
5
xjyj =
j=6
9
aj =
j=n
n+3
Summation Notation
35. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj =
j=6
9
aj =
j=n
n+3
Summation Notation
36. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj =
j=n
n+3
Summation Notation
37. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
38. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in
mathematics.
39. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in
mathematics. An example is the formula for
averaging.
40. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in
mathematics. An example is the formula for
averaging. Given n numbers, f1, f2, f3,.., fn, their
average (mean), written as f,
is (f1 + f2 + f3 ... + fn-1 + fn)/n.
41. Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in
mathematics. An example is the formula for
averaging. Given n numbers, f1, f2, f3,.., fn, their
average (mean), written as f,
is (f1 + f2 + f3 ... + fn-1 + fn)/n. In notation, f =
k=1
n
fk
n
42. The index variable is also used as the variable that
generates the numbers to be summed.
Summation Notation
43. Example D:
a. (k2 – 1)
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
Summation Notation
44. Example D:
a. (k2 – 1) =
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
45. Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
46. Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
47. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
Example D:
48. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2)
k=3
5
Example D:
49. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2)
=(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)k=3
5
Example D:
50. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2)
=(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
= -11 + 14 – 17
k=3
5
Example D:
51. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2)
=(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
= -11 + 14 – 17
= -14
k=3
5
Example D:
52. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2)
=(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
= -11 + 14 – 17
= -14
k=3
5
In part b, the multiple (-1)k change the sums to an
alternating sum, t
Example D:
53. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
= 170
k=5
8
The index variable is also used as the variable that
generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2)
=(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
= -11 + 14 – 17
= -14
k=3
5
In part b, the multiple (-1)k change the sums to an
alternating sum, that is, a sum where the terms
alternate between positive and negative numbers.
Example D:
54. Exercise A. List the first four terms of each of the
following sequences given by fn where n = 1,2, 3, ..
Sequences
2.
4.
5.
6.
7.
8.
9.
10.
1.
3.
fn = –5 + n
fn = 5 – n
fn = 3n
fn = –5 + 2n
fn = 5 – n2
fn = –4n + 1
fn = –5 + nfn = (–1)n5 / n
fn = (3n + 2)/(–1 – n)
fn = 2n2 – n
fn = n2 / (2n + 1)
55. Exercise B. Find a formula fn for each of the
following sequences.
Sequences
12.
14.
15.
16.
17.
18.
19.
20.
11.
13.
2, 3, 4, 5..
–3, –2, –1, 0, 1..
10, 20, 30, 40,..
5, 10, 15, 20,..
–40, –30, –20, –10, 0,..
–5, –10, –15, –20,..
1/2, 1/3, 1/4, 1/5..
1/2, –2/3, 3/4, –4/5..
–1, 1/4, –1/9, 1/16, –1/25,..
1, r1, r2, r3,..
