2. Sequences & Summation Notation 8.1 JMerrill, 2007 Revised 2008

3. Sequences In Elementary School… 12 32

4. And… 17 12

5. Even 22

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

7. 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.

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

9. 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

10. 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

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

12. 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 )…

13. Finding the n th 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

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

15. 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.

16. Example If an = an-1 + 4 and a1 = 3, give the first five terms of the sequence. a1 = 3 If n = 2: a2 = a1 + 4 = 3 + 4 = 7 If n = 3: a3 = a2 + 4 = 7 + 4 = 11 If n = 4: a4 = a3 + 4 = 11 + 4 = 15 If n = 5: a 5 = a4 + 4 = 15 + 4 = 19

17. 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

18. Example Write the first 4 terms of the sequence a0 = 1 a1 = 2 a2 = 2 a3 = 4/3 a4 = 2/3

19. 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

20. Example Write the first four terms of the sequence

21. Evaluating Factorials in Fractions Evaluate:

22. 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

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

24. 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 k 2 for values of k from 1 to 100.” can be read as “the sum of k 2 for values of k from 1 to 100.”

25. Definition of a Series Consider the infinite series a 1, a 2, … a n … 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

26. 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.

27. Definitions Summand Index of Summation Limits of Summation

28. Example

29. Sigma Notation Representing Infinite Series

30. Give the series in expanded form: 5+10+15+20

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

32. One More: Find the Sum of 1089

33. Properties of Sums

34. Last Problem Find the sum of