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Precalculus

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Title: Precalculus


1
Precalculus MAT 129
  • Instructor Rachel Graham
  • Location BETTS Rm. 107
  • Time 8 1120 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.
  • Dont 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)k2k.

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 an 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
  • Sn 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 150th 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
  • an a1rn-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.
  • nPr 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
  • nCr 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?
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