Precalculus

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Precalculus MAT 129

• Instructor Rachel Graham
• Location BETTS Rm. 107
• Time 8 1120 a.m. MWF

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Chapter Eight

• Sequences, Series, and Probability

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Ch. 8 Overview

• Sequences and Series
• Arithmetic Sequences and Partial Sums
• Geometric Sequences and Series
• Counting Principles
• Probability

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8.1 Sequences and Series

• Sequences
• Factorial Notation
• Summation Notation
• Series

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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.

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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.

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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.

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Solution Example 2.8.1

• Pg. 587 11
• a1 0
• a2 1
• a3 0
• a4 ½
• a5 0

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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.

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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.

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Activities (583)

• 1. (2n 2)!/(2n 4)!
• 2. 2n!/4n!
• 3. (2n 1)!/(2n)!

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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!!

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Example 4.8.1

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

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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.

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Example 5.8.1

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

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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.

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8.2 Arithmetic Sequences and Partial Sums

• Arithmetic Sequences
• The Sum of a Finite Arithmetic Sequence

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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.

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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.

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Example 1.8.2

• Pg. 592 Example 1
• These are examples of arithmetic sequences.

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Example 2.8.2

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

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Example 3.8.2

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

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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).

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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.

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Example 5.8.2

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

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8.3 Geometric Sequences and Series

• Geometric Sequences
• Geometric Series
• Application

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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

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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, .

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8.3 Geometric Series

• The sum of the terms of a finite geometric
  sequence is called a geometric sequence.

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8.6 Counting Principles

• Simple Counting Problems
• Permutations
• Combinations

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8.6 Simple Counting Problems

• Many simple counting problems consist of adding
  the total number of possible outcomes.

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Example 1.8.6

• Pg. 627 Examples 1 2
• If you need more practice try Exercise 7.

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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)!

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Example 2.8.6

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

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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!

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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?