Ch. 9 Sequences, Series and Probability

9.1 Sequences and Series

A. OBJ: to find the sum of a series, to use sequence notation, to use factorial notation and to use series notation.

B. FACTS/FORMULAS:

1. sequence means that it is ordered so that it has a first member, a second member, a third member, and so on. Mathematically, you can think of a sequence as a function whose domain is the set of positive integers.

2. sequences are usually written using subscript notation

a1, a2, a3…….

3.

4. factorial – special type of product

5. Notice that 0! = 1 and 1! = 1.

6. 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 .

7.

8. Series - the sum of the terms of a finite or

infinite sequence.

9.2 Arithmetic Sequences and Partial Sums

A. OBJ: to write and find the nth terms of an arithmetic sequence, to find the nth partial sum of an arithmetic sequence.

B. FACTS/FORMULAS:

1. arithmetic sequence - A sequence whose consecutive terms have a common difference.

2.

3. recursive formula - find any term of an arithmetic sequence, provided that you know the preceding term.

an + 1 = an + d.

4.

5. The sum of the first n terms of an infinite sequence is the n th partial sum.

The n th partial sum can be found by using the formula for the sum of a finite arithmetic sequence.

9.3 Geometric Series

A. OBJ: to write and find the terms of a geometric series, to find the sum of an infinite and finite geometric series.

B. FACTS/FORMULAS:

1. geometric series - Consecutive terms that have a common ratio.

2.

3.

4.

5.

** Note that if | r | 1, the series does not have a sum.

6. recursive formula – an = r*an-1

9.5 Binomial Theorem

A. OBJ: to use binomial thm and Pascal’s triangle to calculate the binomial coefficients, to write binomial expansions.

B. FACTS/FORMULAS:

1. binomial is a polynomial that has two terms.

2. binomial coefficients using binomial theorem

3. Pascal’s triangle

4. binomial expansion - nCr xn – r y r.

(x + y)0 = 1

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y 3

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y 4

(x + y)5 = x5 + 5x 4y + 10x3y2 + 10x2y 3 + 5xy 4 + y 5

9.6 Counting Principals

A. OBJ: to solve counting problems using the Counting Principal, Permutation or Combination

B. FACTS/FORMULAS:

1. Make a list to describe simple counting problem which has each possible way that an event can occur.

2.

This can also be used for more than 2 events.

3. When counting and things need to be in a certain order:

4.

5.

6.

7. When counting and the order does not matter:

9.7 Probability

A. OBJ: to find the probabilities of events, mutually exclusive events, independent events, and the complement of an event.

B. FACTS/FORMULAS:

1. Any happening for which the result is uncertain is called an

experiment.

2. The possible results of the experiment are outcomes.

3. The set of all possible outcomes of the experiment is the sample space of the experiment.

4. Any subcollection of a sample space is an event.

5.

6. Because the number of outcomes in an event must be less than or equal to the number of outcomes in the sample space, the probability of an event must be a number between 0 and 1.

0 P(E) 1

7. If P(E) = 0, then event E cannot occur, and E is called an

impossible event.

8. If P(E) = 1, event E must occur, and E is called a certain event.

9. Two events A and B (from the same sample space) are

mutually exclusive when A and B have no outcomes in

common. In the terminology of sets, the intersection of A and B is the empty set, which implies that P(A ∩ B) = 0.

10.

11. Two events are independent when the occurrence of one has no effect on the occurrence of the other. To find the probability that two independent events will occur, multiply the probabilities of each.

12.

13. The complement of an event A is the collection of all outcomes in the sample space that are not in A.

The complement of event A is denoted by A.

Because P(A or A) = 1 and because A and A are mutually exclusive, it follows that P(A) + P(A) = 1.

So, the probability of A is P(A) = 1 – P(A).

14.