The document focuses on summarizing the concepts of sequences and series, including the differences between them and types of sequences such as Fibonacci, arithmetic, geometric, and harmonic. It explains the use of sigma notation for representing series and includes examples for finding sums and associated series. Additionally, it presents problems related to arithmetic and geometric sequences with practical applications of summation notation.

1. The
Summation
Notation

2. Objectives:
• Illustrate a series
• Differentiate a series from a
sequence
• Use the Sigma Notation to
represent a series
• Apply the Sigma Notation in
finding sums

3. Review of Sequences and Series
What are the different types of
sequences and series?

4. Review of Sequences and Series
Types of sequences and Series:
o Fibonacci Sequence
o Arithmetic sequence and
series
o Geometric sequence and
series
o Harmonic series

5. Identify if it is a sequence or a series and
determine its type.
1. 1, 2, 3, …, 10
2. 1+2+3+4+…+10
3. 2, 4, 6, 8, 10
4. 1, ½, 1/3, ¼, …, 1/10
5. 1,1,2,3,5,8,13,21, …

6. Review of Sequences and Series
A sequence is a function whose
domain is the set of positive
integers or the set {1, 2, 3, …, n}

7. Review of Sequences and Series
A series represents the sum of
the terms of a sequence.
Example: 1+2+3+4+…+n

8. Review of Sequences and Series
What’s the difference between a
sequence and a series?

9. Review of Sequences and Series
What’s the difference between a sequence and a
series?
A sequence is a list of numbers (separated by
commas), while a series is a sum of numbers
(separated by “+” or “-” sign)
1, -1/2, 1/3, -1/4 is a sequence
1- ½ + 1/3 – ¼ is a series

10. Types of Sequences and Series
The sequence defined by for , where , is called
a Fibonacci sequence. Its terms are 1, 1, 2, 3, 5,
8, 13, 21, …..

11. Types of Sequences and Series
An arithmetic sequence is a sequence in which each
term after the first is obtained by adding a constant
( common difference) to the preceding term.
If the nth term is , and the common difference is d, then
The associated series with n terms is given by

12. Types of Sequences and Series
A geometric sequence is a sequence in which each
term after the first is obtained by multiplying the
preceding term by a constant (common ratio)
If the nth term is , and the common ratio is r, then
The associated series with n terms is given by
, if r
, if r

13. Types of Sequences and Series
If is an arithmetic sequence, then the sequence
with the nth term is a harmonic sequence.

14. Examples
D e t e r m i n e t h e fi r s t 5 t e r m s o f e a c h d e fi n e d
s e q u e n c e a n d g i v e t h e i r a s s o c i a t e d s e r i e s .

15. Examples
D e t e r m i n e t h e fi r s t 5 t e r m s o f e a c h d e fi n e d
s e q u e n c e a n d g i v e t h e i r a s s o c i a t e d s e r i e s .
F i r s t 5 t e r m s : 1 , 0 , - 1 , - 2 , - 3
A s s o c i a t e d s e r i e s : 1 + 0 - 1 - 2 - 3 = - 5
S o l u t i o n :

16. Examples
D e t e r m i n e t h e fi r s t 5 t e r m s o f e a c h d e fi n e d
s e q u e n c e a n d g i v e t h e i r a s s o c i a t e d s e r i e s .
( 2 )
F i r s t 5 t e r m s : 6 , 1 7 , 3 4 , 5 7 , 8 6
A s s o c i a t e d s e r i e s : 6 + 1 7 + 3 4 + 5 7 + 8 6 = 2 0 0
S o l u t i o n :

17. Examples
D e t e r m i n e t h e fi r s t 5 t e r m s o f e a c h d e fi n e d
s e q u e n c e a n d g i v e t h e i r a s s o c i a t e d s e r i e s .
( 3 )
F i r s t 5 t e r m s : - 1 , 1 , - 1 , 1 , - 1
A s s o c i a t e d s e r i e s : - 1 + 1 - 1 + 1 - 1 = - 1
S o l u t i o n :

18. Examples
D e t e r m i n e t h e fi r s t 5 t e r m s o f e a c h d e fi n e d
s e q u e n c e a n d g i v e t h e i r a s s o c i a t e d s e r i e s .
( 4 )
F i r s t 5 t e r m s : 1 , 3 , 6 , 1 0 , 1 5
A s s o c i a t e d s e r i e s : 1 + 3 + 6 + 1 0 + 1 5 = 3 5
S o l u t i o n :

19. Try this!
1. H o w m a n y t e r m s a r e t h e r e i n a n a r i t h m e t i c
s e q u e n c e w i t h t h e fi r s t t e r m 5 , c o m m o n
d i ff e r e n c e - 3 a n d l a s t t e r m - 7 6 ?
2. L i s t t h e fi r s t t h r e e t e r m s o f a n a r i t h m e t i c
s e q u e n c e i f t h e 2 5 t h
t e r m i s 3 5 a n d t h e 3 0 t h
t e r m i s 5 .
3. T h e s e v e n t h t e r m o f a g e o m e t r i c s e q u e n c e
i s 6 a n d t h e t e n t h t e r m i s 1 6 2 . fi n d t h e fi f t h
t e r m .
4. A b a l l d r o p p e d f r o m t h e t o p o f a b u i l d i n g
1 8 0 m h i g h a l w a y s r e b o u n d s t h r e e - f o u r t h s
t h e d i s t a n c e i t h a s f a l l e n . H o w f a r ( u p a n d
d o w n ) w i l l t h e b a l l h a v e t r a v e l l e d w h e n i t
h i t s t h e g r o u n d f o r t h e 6 t h
t i m e ?

20. The
Summation
Notation

21. What is Sigma Notation?
Let be an expression involving an integer . The
expression
Can be compactly written in a Sigma Notation, and we
write it as
Which is read “the summation of f(i) from i=m to n”. Here,
m and n are integers with mn, is a term (summand) of the
summation, and the letter I is the index, m the lower
bound, and n the upper bound.

22. E x a m p l e s
E x p a n d e a c h s u m m a t i o n , a n d s i m p l i f y i f p o s s i b l e .
2.
3.

23. E x a m p l e s
E x p a n d e a c h s u m m a t i o n , a n d s i m p l i f y i f p o s s i b l e .
2.
3.

24. E x a m p l e s
W r i t e e a c h e x p r e s s i o n i n s i g m a n o t a t i o n
1.
2.
3.
4.

25. Useful Summation Facts

26. Properties of Sigma Notation

27. Properties of Sigma Notation
Telescoping Sum

28. Examples (Properties)
1. Evaluate
2. Derive the formula for using a telescoping sum with
terms