1. Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF

2. Chapter Eight Sequences, Series, and Probability

3. Ch. 8 Overview Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series Counting Principles Probability

4. 8.1 – Sequences and Series Sequences Factorial Notation Summation Notation Series

5. 8.1 – Sequences An infinite sequence is a function whose domain is the set of positive integers. The function values a1, a2, …, an, …. are the terms of the sequence. –If the domain consists only of the first n positive integers only, the sequence is a finite sequence.

6. Example 1.8.1 Pg. 581 Example 2 Write the first five terms of the sequence given by: (-1) n /(2n-1). The algebraic solution only.

7. Example 2.8.1 Pg. 587 #11 Write the first five terms of the sequence given by: (1+(-1) n )/n. Assume n begins with 1.

8. Solution Example 2.8.1 Pg. 587 #11 a1 = 0 a2 = 1 a3 = 0 a4 = ½ a5 = 0

9. 8.1 – Factorial Notation n! = n x (n-1) x … x 3 x 2 x 1. –Called n factorial. –Note the special case 0! = 1.

10. Example 3.8.1 Pg. 583 Example 6 Evaluate the factorial expressions. Do the first one by hand and the last two on your calculator. The factorial key is in the math button menu for most of you.

11. Activities (583) 1. (2n + 2)!/(2n + 4)! 2. 2n!/4n! 3. (2n + 1)!/(2n)!

12. 8.1 – Summation Notation All the summation sign means is that you are going to perform the operations for as long as the limits tell you and then you are going to add them all up. –Don’t let the symbol scare you!!

13. Example 4.8.1 Pg. 584 Example 7 Then see the properties of sums on in the blue box on pg. 561.

14. 8.1 – Series A finite series is a partial sum. –That means that you sum to a number. An infinite series is easy to tell from a finite one because you sum to infinity.

15. Example 5.8.1 Pg. 585 Example 8 Notice that the sum of an infinite series can be a finite number!

16. Activities (585) 1. Write the first five terms of the sequence (assume n begins with 1): (2n - 1)/(2n) 2. Find the sum. from k = 1 to 4 ∑ (-1) k 2k.

17. 8.2 – Arithmetic Sequences and Partial Sums Arithmetic Sequences The Sum of a Finite Arithmetic Sequence

18. 8.2 – Arithmetic Sequences A sequence is arithmetic if the differences between consecutive terms are the same. This means that a2 – a1 = a3 – a2 = a4 – a3 = d.

19. 8.2 – Arithmetic Sequences The number d is the common difference of the arithmetic sequence. The nth term of an arithmetic sequence has the form a n = dn + c.

20. Example 1.8.2 Pg. 592 Example 1 These are examples of arithmetic sequences.

21. Example 2.8.2 Pg. 593 Example 2 Look at the alternative way to find the formula at the bottom of pg. 593.

22. Example 3.8.2 Pg. 594 Example 4 Find the seventh term of the arithmetic sequence whose first two terms are 2 and 9.

23. 8.2 – The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by S n = n/2(a1 + an).

24. Example 4.8.2 Pg. 595 Example 5 Find the sums. For these we need the first term the nth term and the number of terms in the finite sequence.

25. Example 5.8.2 Pg. 596 Example 6 Find the 150 th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, …

26. 8.3 – Geometric Sequences and Series Geometric Sequences Geometric Series Application

27. 8.3 – Geometric Sequences A geometric sequence is one where the ratios of consecutive terms is the same. This common ratio is denoted by the letter r. The nth term of a geometric sequence: a n = a 1 r n-1

28. Example 1.8.3 Pg. 603 Example 4 Find the formula for the nth term of the following geometric sequence. What is the ninth term of the sequence? 5, 15, 45, ….

29. 8.3 – Geometric Series The sum of the terms of a finite geometric sequence is called a geometric sequence.

30. 8.6 – Counting Principles Simple Counting Problems Permutations Combinations

31. 8.6 – Simple Counting Problems Many simple counting problems consist of adding the total number of possible outcomes.

32. Example 1.8.6 Pg. 627 Examples 1 & 2 If you need more practice try Exercise 7.

33. 8.6 – Permutations A permutation of n different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on. n P r = n! / (n - r)!

34. Example 2.8.6 Pg. 630 Example 6 We will use our calculator so you will not have to chug the formula.

35. 8.6 – Combinations These are used when selecting subset of a larger set in which order is not important. Combinations of n Elements Taken r at a Time: n C r = n! / (n - r)!r!

36. Example 3.8.6 Pg. 633 Example 9 A standard poker hand consists of five cards dealt from a deck of 52. How many different poker hands are possible?