1. Sequences & Summation Notation
JMerrill

2. Sequences In Elementary School…12 12 32

3. And… 17 12

4. Even 12 22

5. Sequences
SEQUENCE - a set of numbers, called terms, arranged in a particular order.

6. Sequences
An infinite sequence is a function whose domain is the set of positive integers. The function values a1, a2, a3, …, an… are the terms of the sequence.
If the domain of the sequence consists of the first n positive integers only, the sequence is a finite sequence.
n is the term number.

7. Examples
Finite sequence: 2, 6, 10, 14
Infinite sequence:

8. Writing the Terms of a Sequence
Write the first 4 terms of the sequence an = 3n – 2
a1 = 3(1) – 2 = 1
a2 = 3(2) – 2 = 4
a3 = 3(3) – 2 = 7
a4 = 3(4) – 2 = 10
Calculator steps in LIST

9. Writing the Terms of a Sequence
Write the first 4 terms of the sequence an = 3 + (-1)n
a1 = 3 + (-1)1 = 2
a2 = 3 + (-1)2 = 4
a3 = 3 + (-1)3 = 2
a4 = 3 + (-1)4 = 4

10. You Do
Write the first 4 terms of the sequence

11. Graphs Consider the infinite sequence
Because a sequence is a function whose domain is the set of positive integers, the graph of a sequence is a set of distinct points.
The first term is ½ , the 2nd term is ¼ …
So, the ordered pairs are (1, ½ ), (2, ¼ )…

12. Finding the nth Term of a Sequence
Write an expression for the nth term (an) of the sequence 1, 3, 5, 7…
n: 1, 2, 3, 4…n
Terms: 1, 3, 5, 7…an
Apparent pattern: each term is 1 less than twice n. So, the apparent nth term is
an = 2n - 1
Always compare the term to the term number

13. Finding the nth Term of a Sequence You Do
Write an expression for the nth term (an) of the sequence
Apparent pattern:
n = 1, 2, 3, 4…n
The numerator is 1; the denominator is the square of n.

14. Recursive Definition
Sometimes a sequence is defined by giving the value of an in terms of the preceding term, an-1. A recursive sequence consists of 2 parts:
An initial condition that tells where the sequence starts.
A recursive equation (or formula) that tells how many terms in the sequence are related to the preceding term.

15. Example
If an = an and a1 = 3, give the first five terms of the sequence.
a1 = 3
If n = 2: a2 = a1 + 4 = = 7
If n = 3: a3 = a2 + 4 = = 11
If n = 4: a4 = a3 + 4 = = 15
If n = 5: a5 = a4 + 4 = = 19

16. A Famous Recursive Sequence
The Fibonacci Sequence is very well known because it appears in nature.
The sequence is 1, 1, 2, 3, 5, 8, 13…
Apparent pattern?
Each term is the sum of the preceding 2 terms
The nth term is
an = an-2 + an-1

17. Example Write the first 4 terms of the sequence a0 = 1 a1 = 2 a2 = 2

18. Factorial Notation
Products of consecutive positive integers occur quite often in sequences. These products can be expressed in factorial notation:
1! = 1
2! = 2 ● 1 = 2
3! = 3 ●2 ●1 = 6
4! = 4 ●3 ●2 ●1 = 24
5! = 5 ●4 ●3 ●2 ●1 = 120
The factorial key can be found in MATH PRB:4 on your calculator
0!, by definition, = 1

19. Example
Write the first four terms of the sequence

20. Evaluating Factorials in Fractions
Evaluate:

21. Definitions
The words sequences and series are often used interchangeably in everyday conversation. (A person may refer to a sequence of events or a series of events.) In mathematics, they are very different.
Sequence: a set of numbers, terms, arranged in a particular order
Series: the sum of a sequence

22. Examples Finite sequence: 2, 6, 10, 14 Finite series: 2 + 6 + 10 + 14
Infinite sequence:
Infinite series:

23. Intro to Sigma
The Greek letter (sigma) is often used in mathematics to represent a sum (series) in abbreviated form.
Example: which
can be read as “the sum of k2 for values of k from 1 to 100.”

24. Definition of a Series Consider the infinite series a1, a2, … an…
The sum of the first n terms is a finite series (or partial sum) and is denoted by
The sum of all terms of an infinite sequence is called an infinite series and is denoted by

25. Sigma Continued Similarly, the symbol is read “the
sum of 3k for values of k from 5 to 10.”
This means that the symbol represents the series whose terms are obtained by evaluating 3k for k = 5, k = 6, and so on, to k = 10.

26. Definitions
Limits of Summation
Summand
Index of Summation

27. Example

28. Sigma Notation Representing Infinite Series

29. Give the series in expanded form:

30. Find the Sum of
190
Calculator steps: in LIST

31. One More: Find the Sum of
1089

32. Properties of Sums

33. Last Problem
Find the sum of