1 warm up Find the angle between the two vectors u =  1, 5  v =  4, -3 

2 3 rd 6 weeks Three units left Sequences, series, probability Analytic Geometry Limits Each unit has 1 test, no quizzes

3 Sequences In mathematics, the word sequence is used in much the same way as in ordinary English. Two examples are 1, 2, 3, 4,... and 1, 3, 5, 7,....

4 Sequences Rather than using function notation, however, sequences are usually written using subscript notation, as indicated in the following definition.

5 Sequences On occasion it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become a 0, a 1, a 2, a 3,.... When this is the case, the domain includes 0.

6 Example 1(a) – Writing the Terms of a Sequence The first four terms of the sequence given by a n = 3n – 2 are a 1 = 3(1) – 2= a 2 = 3(2) – 2= a 3 = 3(3) – 2= a 4 = 3(4) – 2=

7 Example 1(b) – Writing the Terms of a Sequence The first four terms of the sequence given by a n = 3 + (–1) n are cont’d

8 Example 1(b) – Writing the Terms of a Sequence The first four terms of the sequence given by a n = a 1 = a 2 = a 3 = a 4 = cont’d

9 Sequences Simply listing the first few terms is not sufficient to define a unique sequence—the n th term must be given. To see this, consider the following sequences, both of which have the same first three terms.

10 Sequences Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. All other terms of the sequence are then defined using previous terms. * recursively just refers to using the previous term to find the next term

11 recursive sequence a 1 = 3, a k = 2(a k-1 ) + 1 Find the first 4 terms of the sequence

12 recursive sequence a 1 = 2, a k = (a k-1 ) - 3 Find the first 4 terms of the sequence

13 recursive sequence a 1 = 4, a k = 3(a k-1 ) + 1 Find the first 4 terms of the sequence

14 Factorial Notation Some very important sequences in mathematics involve terms that are defined with special types of products called factorials.

15 Factorial Notation Notice that 0! = 1 and 1! = 1. Here are some other values of n!. 2! = 1  2 = 2 3! = 1  2  3 = 6 4! = 1  2  3  4 = 24

16 Factorial Notation Factorials follow the same conventions for order of operations as do exponents. For instance, 2n! = 2(n!) = 2(1  2  3  4... n) whereas (2n)! = 1  2  3  n.

17 Example 6 – Writing the Terms of a Sequence Involving Factorials Write the first five terms of the sequence given by Begin with n = 0.

18 Example 6 – Writing the Terms of a Sequence Involving Factorials Write the first five terms of the sequence given by Begin with n = 0.

19 Factorial Notation When working with fractions involving factorials, you will often be able to reduce the fractions to simplify the computations.

20 Factorial Notation

21 Summation Notation A convenient notation for the sum of the terms of a finite sequence is called summation notation or sigma notation. It involves the use of the uppercase Greek letter sigma, written as .

22 Example 8 – Summation Notation for a Sum a. = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) b. =

23 Example 8 – Summation Notation for a Sum

24 summation on the calculator

25 summation in calculator**

26 Summation Notation **Important in calculus

27 Series Many applications involve the sum of the terms of a finite or infinite sequence. Such a sum is called a series.

28 Example 9 – Finding the Sum of a Series For the series find (a) the third partial sum (b) the sum.

29 Example 10 – Compound Interest An investor deposits $5000 in an account that earns 3% interest compounded quarterly. The balance in the account after n quarters is given by a. Write the first three terms of the sequence. b. Find the balance in the account after 10 years by computing the 40th term of the sequence.

30 Homework Page 613 7,11, 23, 25 Page , 51, (odd)