The document discusses sequences and summation notation. It defines a sequence as an ordered list of numbers that may have a pattern. Common examples provided are the sequences of odd numbers, even numbers, and square numbers. A formula is given for calculating the nth term of each sequence. Summation notation is introduced as using the Greek letter sigma to represent summing a list of numbers. An example shows how to write the sum of 100 terms in a sequence using sigma notation with limits and an index variable.

1. Sequences

2. Sequences
Here is a typical spread sheet that
records an ordered list of numbers
under the column F.

3. Sequences
Here is a typical spread sheet that
records an ordered list of numbers
under the column F.
We may view the entries 2, 4, 6…
as guests in rooms F1, F2, F3 …
respectively, in a hotel.

4. Sequences
Here is a typical spread sheet that
records an ordered list of numbers
under the column F.
We may view the entries 2, 4, 6…
as guests in rooms F1, F2, F3 …
respectively, in a hotel.
By ordered list we mean that
if the entries were 4, 2, 6, 8,10, …
we would treat it as a different list.

5. Sequences
Here is a typical spread sheet that
records an ordered list of numbers
under the column F.
We may view the entries 2, 4, 6…
as guests in rooms F1, F2, F3 …
respectively, in a hotel.
By ordered list we mean that
if the entries were 4, 2, 6, 8,10, …
we would treat it as a different list.
The entries 2, 4, 6.. is the list of even numbers.
Hence we may infer that F10 = 20.
In fact, there is a formula for calculating the entry FN
based on the “room number” N that FN = 2N.

6. A sequence is an ordered list of infinitely many
numbers that may or may not have a pattern.
Sequences

7. 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.

8. 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.

9. 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.

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.

11. 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, …

12. 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, … so that
f100 = 100th number on the list,

13. 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, … so that
f100 = 100th number on the list,
fn = the n’th number on the list,

14. 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, … so that
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
right before fn.

15. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that f3 = 9, f4= 16, f5 = 25,
Sequences

16. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that f3 = 9, f4= 16, f5 = 25,
and a formula for fn is fn = n2.
Sequences

17. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that 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, …
we have that f3= 6, f4 = 8, f5 = 10,
Sequences

18. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that 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, …
we have that f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2n
Sequences

19. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that 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, …
we have that f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2n
Sequences
c. A formula for the sequence of odd numbers
1, 3, 5,.. is fn = 2n – 1.

20. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that 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, …
we have that f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2n
Sequences
c. A formula for the sequence of odd numbers
1, 3, 5,.. is fn = 2n – 1.
d. Multiplying (–1)n or n ±1 creates a sequence with
alternating signs,

21. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that 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, …
we have that f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2n
Sequences
c. A formula for the sequence of odd numbers
1, 3, 5,.. is fn = 2n – 1.
d. Multiplying (–1)n or n ±1 creates a sequence with
alternating signs, e.g. fn = (–1)n(2n – 1)
gives –1, 3, –5, 7,.., the odd numbers with
alternating ± signs.

22. Example B.
a. For the sequence of square numbers 1, 4, 9, 16, …
we have that 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, …
we have that f3= 6, f4 = 8, f5 = 10,
and a formula for fn is fn = 2n
Sequences
c. A formula for the sequence of odd numbers
1, 3, 5,.. is fn = 2n – 1.
d. Multiplying (–1)n or n ±1 creates a sequence with
alternating signs, e.g. fn = (–1)n(2n – 1)
gives –1, 3, –5, 7,.., the odd numbers with
alternating ± signs. Such a sequence whose
signs alternate is called an alternating sequence.

23. Summation Notation

24. In mathematics, the Greek letter “” (sigma) means
“to sum”.
Summation Notation

25. In mathematics, the Greek letter “” (sigma) means
“to sum”. Hence, “x” means to add the x’s,
Summation Notation

26. In mathematics, the Greek letter “” (sigma) means
“to sum”. Hence, “x” means to add the x’s,
“(x*y)” means to add the x*y’s.
Summation Notation

27. In mathematics, the Greek letter “” (sigma) means
“to sum”. 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

28. Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
In mathematics, the Greek letter “” (sigma) means
“to sum”. 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
In mathematics, the Greek letter “” (sigma) means
“to sum”. 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.
In mathematics, the Greek letter “” (sigma) means
“to sum”. 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:
 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 sum”. 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:
 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 sum”. 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.)

33. 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 sum”. 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. 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 sum”. 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.)

35. 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 sum”. 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.)

36. Example C.
 fk =k=4
8
 ai =
i=2
5
 xjyj =
j=6
9
 aj =
j=n
n+3
Summation Notation

37. 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

38. 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

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

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 is used to express formulas.

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 is used to express formulas.
An example is the formula for the mean (average).

42. 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 is used to express formulas.
An example is the formula for the mean (average).
Given n numbers, f1, f2, f3,.., fn, their mean or average,
written as f

43. 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 is used to express formulas.
An example is the formula for the mean (average).
Given n numbers, f1, f2, f3,.., fn, their mean or average,
written as f = (f1 + f2 + f3 ... + fn–1 + fn)/n.

44. 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 is used to express formulas.
An example is the formula for the mean (average).
Given n numbers, f1, f2, f3,.., fn, their mean or average,
written as f = (f1 + f2 + f3 ... + fn–1 + fn)/n.
f =
k=1
n
fk
nIn  notation, .

45. The index variable is also used as the variable that
generates the numbers to be summed.
Summation Notation

46. 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

47. 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

48. 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

49. 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

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
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)
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)
k=3
5
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
k=3
5
Example D.

54. 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.

55. 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.

56. Properties Summation Notation
–Notation Properties

57. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
–Notation Properties
The –notation distributes over sums and differences.
b.  (ak – bk) =  ak –  bk

58. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk

59. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk

60. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk
for example,  3k=1
5

61. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk
for example,  3 = 3 + 3 + 3 + 3 + 3 = 15.k=1
5

62. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk
for example,  3 = 3 + 3 + 3 + 3 + 3 = 15.k=1
5
To verify these rules write down the sums, e.g.
 (ak + bk)k
n

63. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
 (ak + bk)
= (a1 + b1) + (a2 + b2) + .. + (an + bn)
k
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk
for example,  3 = 3 + 3 + 3 + 3 + 3 = 15.k=1
5
To verify these rules write down the sums, e.g.

64. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
 (ak + bk)
= (a1 + b1) + (a2 + b2) + .. + (an + bn)
= (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)
k
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk
for example,  3 = 3 + 3 + 3 + 3 + 3 = 15.k=1
5
To verify these rules write down the sums, e.g.

65. Properties Summation Notation
a.  (ak + bk) =  ak +  bk
c.  cak = c(ak) where c is a constant.
d. If c is a constant, then  c = c + c + .. + c = nc.
k=1
n times
n
 (ak + bk)
= (a1 + b1) + (a2 + b2) + .. + (an + bn)
= (a1 + a2 + .. + an) + (b1 + b2 + .. + bn) =
k
 ak +k
 bk.k
n
–Notation Properties
The –notation distributes over sums and differences.
Constant multiples may be pulled out.
b.  (ak – bk) =  ak –  bk
for example,  3 = 3 + 3 + 3 + 3 + 3 = 15.k=1
5
To verify these rules write down the sums, e.g.

66. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:

67. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:
Sn = 1 + 2 + ……+ (n – 1) + n
Sn = n + (n – 1) + ……+ 2 + 1+

68. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:
Sn = 1 + 2 + ……+ (n – 1) + n
Sn = n + (n – 1) + ……+ 2 + 1
2Sn=(n+1)+(n+1)+……+(n+1)+(n+1)
n times
+

69. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:
Sn = 1 + 2 + ……+ (n – 1) + n
Sn = n + (n – 1) + ……+ 2 + 1
2Sn=(n+1)+(n+1)+……+(n+1)+(n+1)
n times
Hence 2Sn = n(n + 1) or Sn =
n(n + 1)
2
+

70. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:
Sn = 1 + 2 + ……+ (n – 1) + n
Sn = n + (n – 1) + ……+ 2 + 1
2Sn=(n+1)+(n+1)+……+(n+1)+(n+1)
n times
Hence 2Sn = n(n + 1) or Sn =
n(n + 1)
2
 k =
Formula for the Sum of Natural Numbers
k=1
n
n(n + 1)
2
+

71. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:
Sn = 1 + 2 + ……+ (n – 1) + n
Sn = n + (n – 1) + ……+ 2 + 1
2Sn=(n+1)+(n+1)+……+(n+1)+(n+1)
n times
Hence 2Sn = n(n + 1) or Sn =
n(n + 1)
2
For example 1+2+3..+100 =  kk=1
100
 k =
Formula for the Sum of Natural Numbers
k=1
n
n(n + 1)
2
+

72. Properties Summation Notation
The formula for the sum of the first n numbers
1 + 2 + .. + (n – 1) + n = Sn may be found by summing
two copies in the following manner:
Sn = 1 + 2 + ……+ (n – 1) + n
Sn = n + (n – 1) + ……+ 2 + 1
2Sn=(n+1)+(n+1)+……+(n+1)+(n+1)
n times
Hence 2Sn = n(n + 1) or Sn =
n(n + 1)
2
For example 1+2+3..+100 =  k =k=1
100
100(100 + 1)
2
= 5050
 k =
Formula for the Sum of Natural Numbers
k=1
n
n(n + 1)
2
+

73. Properties Summation Notation
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

74. Properties Summation Notation
Example E. Find  (2k – 5)
k=1
45
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

75. Properties Summation Notation
k=1
45
 (2k – 5) = Σ2k – Σ5 by property a
k=1
45
k
Example E. Find  (2k – 5)
k
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

76. Properties Summation Notation
k=1
45
 (2k – 5) = Σ2k – Σ5 by property a
= 2Σk – Σ5 by property c
k=1
45
kk
k=1 k=1
45 45
Example E. Find  (2k – 5)
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

77. Properties Summation Notation
k=1
45
 (2k – 5) = Σ2k – Σ5 by property a
= 2Σk – Σ5 by property c
k=1
45
k
k=1
45
= 2 45(45 + 1)
2
by the sum formula
Example E. Find  (2k – 5)
k
45
k=1
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

78. Properties Summation Notation
k=1
45
 (2k – 5) = Σ2k – Σ5 by property a
= 2Σk – Σ5 by property c
k=1
45
k
k=1
45
= 2 – 5*4545(45 + 1)
2
by the sum formula by property d
Example E. Find  (2k – 5)
k
45
k=1
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

79. Properties Summation Notation
k=1
45
 (2k – 5) = Σ2k – Σ5 by property a
= 2Σk – Σ5 by property c
k=1
45
k
k=1
45
= 2 – 5*4545(45 + 1)
2
by the sum formula by property d
= 2070 – 225
= 1845
Example E. Find  (2k – 5)
k
45
k=1
We may use the above properties and the sum
formula to sum all linear sums of the form  (ak + b).k

80. Properties Summation Notation
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

81. Properties Summation Notation
k=10
53
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.
Example F. Find  (4k – 33)

82. Properties Summation Notation
k=10
53
Select a new index, say m, to start at 1.
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

83. Properties Summation Notation
k=10
53
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

84. Properties Summation Notation
k=10
53
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

85. Properties Summation Notation
k=10
53
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

86. Properties Summation Notation
k=10
53
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

87. Properties Summation Notation
k=10
53
 (4k – 33) =
k=10
53
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Rewriting the sum in terms of m:
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

88. Properties Summation Notation
k=10
53
 (4k – 33) = Σ[4(m + 9) – 33]
k=10
53
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Example F. Find  (4k – 33)
Rewriting the sum in terms of m:
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

89. Properties Summation Notation
k=10
53
 (4k – 33) = Σ[4(m + 9) – 33]
k=10
53
m=1
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Rewriting the sum in terms of m:
44
Example F. Find  (4k – 33)
Adjust the range.
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

90. Properties Summation Notation
k=10
53
 (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3)
k=10
53
m=1
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Rewriting the sum in terms of m:
44
m=1
44
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

91. Properties Summation Notation
k=10
53
 (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3)
= 4Σm – Σ3
k=10
53
m=1
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Rewriting the sum in terms of m:
44
m=1
44
m=1
44
m=1
44
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

92. Properties Summation Notation
k=10
53
 (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3)
= 4Σm – Σ3
k=10
53
m=1
= 4 – 3*4444(44 + 1)
2
= 3828
Select a new index, say m, to start at 1.
The lower numbers are k = 10 and m = 1  k = m + 9.
The upper number is k = 53  53 = m + 9 or m = 44.
Rewriting the sum in terms of m:
44
m=1
44
m=1
44
m=1
44
Example F. Find  (4k – 33)
If the linear sum does not start at k = 1 as required,
use substitution to shift the index-variable to start at 1,
then utilize the sum formula.

93. Recurrent Sequences
Another method for describing sequences is
to give the recurrent-formulas for an based on
the previous terms an–1, an–2, ...etc.
We've 1 pair of baby turtles
at time 0.
It takes 1 year for them to mature.
After the 1st year, they will
produce another baby pair just
like them every year.
Let an = the number of pairs of
turtles at the end of the n’th year.
This is the Fibonacci sequence
which is 1, 1, 2, 3, 5, 8, 13,… etc
It’s defined as a0 = 1, , and that an = an-1 + an-2 .
a0 = 1
a1 = 1
1 yr
baby
new
new
mature
new
new
1 yr
1 yr
1 yr
new
new

94. Recurrent Sequences
Another method for describing sequences is
to give the recurrent-formulas for an based on
the previous terms an–1, an–2, ...etc.
We've 1 pair of baby turtles
at time 0.
It takes 1 year for them to mature.
After the 1st year, they will
produce another baby pair just
like them every year.
Let an = the number of pairs of
turtles at the end of the n’th year.
This is the Fibonacci sequence
which is 1, 1, 2, 3, 5, 8, 13,… etc
It’s defined as a0 = 1, , and that an = an-1 + an-2 .
a0 = 1
a1 = 1
1 yr
baby
new
new
mature
new
new
1 yr
1 yr
1 yr
new
new

95. 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 – n2fn = –4n + 1
fn = (–1)n5 / n
fn = (3n + 2)/(–1 – n)
fn = 2n2 – n
fn = n2 / (2n + 1)
B. Find formulas fn for the following sequences.
2.
4.
5. 6.
7. 8.
9.
10.
1.
3.
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, 0.1, 0.001, 0.0001,..

96. Sequences

97. Sequences
It’s possible to add infinitely many numbers and obtain a
finite sum. For example, the sum ½ + ¼ + 1/8 + 1/16...
represents the accumulated amount of
“taking half of the 1 or ½,
take half of what’s left, or ¼,
then take of half of what’s left or 1/8,
and repeat the process without stopping..”
We see that ½ + ¼ + 1/8 + 1/16 + 1/32... = 1.
..
= 1
7. What is 1/3 + 1/9 + 1/27 + 1/81... = ?
½
¼
1/8
1/16
1/32
(Hint: Let the sum 1/3 + 1/9 + 1/27 + 1/81... = x,
factoring out 1/3 from the left, we’ve
1/3(1 + 1/3 + 1/9 + 1/27 + 1/81...) = x, or
1/3(1 + x) = x, then solve for x.)
8. What is 1/4 + 1/16 + 1/64 + 1/81... = ? (Hint: factor out ¼)
9. What is 1/5 + 1/25 + 1/125 + 1/625... = ? (Hint: factor out 1/5)

98. (Answers to the odd problems) Exercise A.
1. f1 = –4, f2 = –3, f3 = –2, f4 = –1
3. f1 = 3, f2 = 6, f3 = 9, f4 = 12
5. f1 = –3, f2 = –7, f3 = –13, f4 = – 15
9. f1 = -5/2, f2 = -8/3, f3 = -11/4, f4 = -14/5
7. f1 = 1, f2 = 7, f3 = 17, f4 = 31
Exercise B.
1. fn= n+1 3. fn= 10n 5. fn= –10(5–n)
7. fn= 1/(n+1) 9. fn= (–1)n/n2
Exercise C.
1. 16 3. 26 5. 70
9. 60
7. 90
11. 436 13. 11/6 15. 100/101
Sequences

99. Exercise D.
1. 6(𝑘 + 1) 3. 9𝑗 − 3 5. 21(𝑘 + 3)
7. 1/2 9. 1/4
Sequences