Chapter 11: Sequences, Induction, and ProbabilityStudy Guide |
Section 11.1
Hints:
1. An infinite sequence is a function whose domain is the set
of positive integers. The function's value or terms are represented by 
,
where n is the number of the term. So
, is the
first term, 
is the second term and so on.
2. The notation, 
,
represents the nth term or general term
of the sequence. Given a formula for the general term, you can find any
term or terms in the sequence.
3. A recursion formula can also be used to define a
sequence. A recursion formula defines the nth
term of a sequence as a function of the previous term. For
example, if you want to find 
,
you would substitute the value of 
into the recursion formula.
4. Sometimes sequences contains factorials. Factorials are products of consecutive positive integer. A general way to write factorial notation is n!.
Ex.: 
Notice that you multiply 5 by every integer less than 5.
5. The sum of the first n terms of a sequence can
be written in summation notation. Summation notation uses the Greek letter
.
In summation notation,
where i is the index of summation, n is the upper limit, and 1 is the lower limit. You do not have to use i for the index. Any number can be used, but the most common are i, j, and k.
| Additional Exercises: | |
1. |
Q: Find the first four terms of the sequence whose general term is given. |
| 2. |
Q: Find the first four terms of the sequence using the given
recursion formula.
|
| 3. |
Q: Find the indicated sum.
|
Section 8.2
Hints:
1. Arithmetic sequences have a
common difference between their terms (numbers). For example:
1, 4, 7, 10, ..., n has a difference of 3 (three is added to get the next
number), while 1, -2, -5, -8, ...,n has a difference of -3 (three is
subtracted to get the next number). Remember!
To find the common difference you can always subtract
the first term in the sequence from the second term in the sequence, 
.
2. Since an arithmetic sequence is infinite and following the same pattern
throughout, we talk about a nonspecific, general term as the nth term
from the numbers 1, 2, 3, ... n where n is really infinitely
large. The formula 
is used to find the nth term in an arithmetic
sequence. The n can be any counting number. 
is the first number of the sequence, and d is the common difference.
3. To find the sum of n
terms in an arithmetic sequence use the
formula 
.
In order to use this formula, you must know 
and 
. If 
is not
given, you have to use the general term formula given in #2 above and find it first.
Additional Exercises:
1.
Q: Write the first five terms of the arithmetic sequence where a1
= 6 and d = -4.
2.
Q: Find a8 when a1 = -5
and d = 4 for an arithmetic sequence.
3.
Q: Find the sum of the first 20 terms of the sequence 3,
-2, -7, -12. . ..
Section 8.3
Hints:
1. A geometric sequence is a list of numbers where the
next number is generated by multiplying by a constant. This constant is
called r, the common ratio, and
can be positive or negative. Sometimes the ratio is a fraction. For example:
80, 40, 20, 10, ... has a ratio of 
.
Remember! You
can always find the common ratio for a geometric sequence by dividing the
second term in the sequence by the first term in the sequence 
.
2. The formula for the nth
term of a geometric sequence is 
,
where 
is the first term of the sequence and r is the common ratio. It
is very important when using this formula to watch your signs if the ratio
is negative.
3. To find the sum of n
terms in a geometric sequence use the formula, 
.
All that you have to know to use this formula is r and 
.
Be careful when working with this formula if the ratio
is a fraction.
4. If r is between
-1 and 1, then the sum of the infinite
geometric sequence can be found using the formula, 
.
This formula can also be used to change a repeating
decimal into a fraction. You can see an example of this type of
problem on page 956.
| Additional Exercises: | |
| 1. |
Q: What is the ratio, r, for this geometric sequence: 180, 60, 20, ...? |
| 2. |
Q: What is a9 when a1 = 4 and r = -2 for a geometric sequence? |
| 3. |
Q: Find the sum of the first 7 terms of the sequence -3, 12, -48, 192. . .. |
Section 8.4
Hints:
1. Mathematical induction is used to prove statements about positive integers. There are two steps in a proof by mathematical induction.
2. Let 
be a statement involving the positive integer n.
To prove 
is true for all positive integers--
Step 1.
Show that 
true.
Step 2.
Show that if 
is assumed to be true, then 
is also true, for every positive integer k.
3. In Step 2 of the proof, you will need to be very careful. Quite a bit of algebraic simplification may be necessary in this step.
| Additional Exercises: | |
| 1. |
Q: Write 
for the following statement and simplify.
|
| 2. |
Q: Write 
for the following statement and simplify.
|
Section 8.5
Hints:
1. When you write out the binomial expansion
,
where n is a positive integer, you will see a
number of patterns begin to appear. It is helpful to use these
patterns when expanding any binomial.
2. In order to find the coefficients of the
binomial expansion, you can use a formula defined
in terms of n and r. n and r are
nonnegative integers, with 
.
n is the power to which you are raising the
binomial, and r will change from 0 to
n. The notation used for the definition of the binomial coefficient is 
,
which is read " n above r".
3. In order to expand a binomial completely, you use the binomial coefficient formula and a pattern for the variable part of each term. A formula called the Binomial Theorem can be written for any positive integral power of a binomial.
For any positive integer n,
.
4. Remember! When using the Binomial Theorem, the power of the first term in the binomial is always n and will decrease with every term that follows. As this happens, the power of the second term in the binomial begins at zero and increases with every term that follows. A binomial expansion will always have n+1 terms.
5. A particular term in a binomial expansion can be found using the formula below. This formula finds the (r+1)st term. In order to find r, you must solve for it. For example, if you want to find the sixth term, then you would solve the equation r + 1 = 6 for r, and r would be 5.
| Additional Exercises: | |
| 1. |
Q: Expand the binomial. |
| 2. |
Q: Find the fifth term of the binomial expansion. |
Section 8.6
Hints:
_____ _____ _____ _____ _____
and 
| Additional Exercises: | |
| 1. |
Q: You are taking a trip that requires three kinds of transportation: flying, sailing, and driving. In how many ways can you travel, if there are 3 different airlines, 4 different cruise ship companies, and 6 different bus tour companies available for your trip? |
| 2. |
Q: A 12-member dance company is holding auditions for a children's play based on Little Red Riding Hood. In how many different ways can the parts of Little Red Riding Hood, The Wolf, Grandmother, and The Woodsman be chosen? |
| 3. |
Q: A small company wants to send a delegation to a technical conference. If there are 10 employees in the company and only four may go to the conference, how many ways can the selection be made? |
Section 8.7
Hints:
P(E)= number of outcomes in event E number of outcomes in sample space S
| Additional Exercises: | |
| 1. |
Q: Find the probability of drawing a red 10 from a standard 52-card deck. |
| 2. |
Q: One card is drawn from a standard 52-card deck. Find the probability of drawing a red card or a queen. |
| 3. |
Q: The probability of a horse not winning a race is  .
What is the probability of the horse winning the race?
|
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