Download Mathematical Induction

You do ……
Do in calculator, too.
Sequence formulas that give you the exact
definition of a term are explicit formulas.
Formula:
an = 5n
Explicit, every term is
defined by this formula.
Recursive formulas define terms based on
preceding terms.
Formula:
a1 = 1
a2 = 1
an = an −1 + an − 2 , for _ n ≥ 3
Recursive, after the
second term and the
two previous terms to
get the next term.
Formula:
a1 = 1
a2 = 1
an = an −1 + an − 2 , for _ n ≥ 3
Find the first 6 terms of
the sequence.
a1 = 1
a2 = 1
a3 = a2 + a1 = 1 + 1 = 2
a4 = a3 + a2 = 2 + 1 = 3
a5 = a4 + a3 = 3 + 2 = 5
a6 = a5 + a4 = 5 + 3 = 8
You do …..
Problems for you………..
Page 587-588
1,5,9,13,17,21,25,27,31,35,37,
45,51,53
More problems to do…….
Page 588
57,59,61,63,65,67,69,71,73,77
Notation for multiple sums:
∑
sigma
Definition _ of _ Sigma _ Notation:
The _ sum_ of _ n _ terms _ a1 , a2 , a3 ,..., an
is _ written _ as
n
∑a
i
= a1 + a2 + a3 +.....+ an
i =1
i = index _ of _ summation
ai = ith _ term_ of _ the _ sum
lower _ bound = 1
upper _ bound = n
You do ….
Summation Formulas
3
n
∑ 5 = 5 + 5 + 5 = 5(3) = 15
∑ c = cn
i =1
i =1
n(n + 1)(2n + 1)
i =
∑
6
i =1
n
2
n(n + 1)
i=
∑
2
i =1
n
n (n + 1)
i =
∑
4
i =1
n
3
2
4
! i2 =
i=1
(
4(5)(9)
= 30 " 12 + 2 2 + 32 + 4 2 = 30
6
)
3(3 +1)
i=
= 6 " (1 + 2 + 3 = 6 )
!
2
i=1
3
2
2
2
2
(2
+1)
3
3
3
i
=
=
9
"
1
+
2
=9
!
4
i=1
2
(
)
Evaluate _ the _ sum
i +1
∑
2
i =1 n
for _ n = 10_ & _ n = 1000
n
i +1 1 n
= 2 ! (i +1) " by _ Theorem
!
2
n i=1
i=1 n
n
n
%
1 n
1 " n
i +1) = 2 $ ! i + !1' ( by _ Theorem
2 !(
n i=1
n # i=1
i=1 &
n
% 1 " n(n +1)
1 " n
%
i
+
1
=
+
n
( by _ Theorem
!
'
2 $!
2 $
'
n # i=1
n # n
&
i=1 &
Evaluate _ the _ sum
i +1
∑
2
i =1 n
for _ n = 10_ & _ n = 1000
n
n
% 1 " n(n +1)
1 " n
%
i
+
1
=
+
n
( by _ Theorem
!
'
2 $!
2 $
'
n # i=1
n # n
&
i=1 &
1 ! n(n +1)
$ n+3
+
n
=
2 #
&
n " n
2n
%
13
when _ n = 10 '
= .65
20
1003
when _ n = 1000 '
= .50015
2000
You do…..
Problems for you ….
Page 588-589
79-109 odds
You do ….
You do ….
Problems for you……
From page 598
#1-9 odds, 13, 15, 19, 23, 27, 29, 31,
35, 37
Problems for you….
Page 598-599
39-69 odds
You do …
You do….
You do ….
Calculator
List Math 5
List Ops 5
You do …. (by hand and calculator)
Problems for you….
Page 607
1,5,9,13,17,19,23,27,31,33,35
We need to do a cumulative sum for an infinite geometric
Cumsum under LIST OPS
Scroll
right
An example of inductive reasoning
Looking for patterns…
Some of these patterns hold for several early terms but break down
afterwards….
One of the most famous of these is Fermat s
conjecture.
These are all primes. One of the troubles with
evaluating Fermat s conjecture was that the
numbers grow very large very quickly
You do ….
You do….
Problems for you…..
Pg 617
1-11 all
Part 2
You do ….
Let s do the first one
together…….
Showing it works for when n = 1 :
1
∑i =1
i =1
n(n + 1) (1)(2)
=
=1
2
2
check.
Showing if it works for nth case it works for n + 1 case :
n +1
∑1 + 2 + 3 +  + n + (n + 1) =
i =1
n(n + 1)
+ (n + 1)
2
n(n + 1) 2(n + 1) n 2 + n + 2n + 2 n 2 + 3n + 2
=
+
=
=
2
2
2
2
n 2 + 3n + 2 (n + 1)(n + 2)
=
=
2
2
check
Formula for n + 1 case :
(n + 1)((n + 1) + 1) (n + 1)(n + 2)
=
2
2
If you are using Mathematical induction to show that
something holds for n greater than some number, say
for 4, then……………..
1. Show it holds for the case when n = 4
2. Show if it holds for the case n then that it holds for
the case n+1…..(exactly like before)
You do….
Problems for you…..
From pg 617
17-25 odds
You do…..
Note:
You do …..
⎛ 4⎞
⎜⎜ ⎟⎟ = 1
⎝0⎠
(x + 2 )
4
⎛ 4⎞
⎜⎜ ⎟⎟ = 4
⎝1 ⎠
⎛ 4⎞
⎜⎜ ⎟⎟ = 6
⎝ 2⎠
⎛ 4⎞
⎜⎜ ⎟⎟ = 4
⎝3⎠
⎛ 4⎞
⎜⎜ ⎟⎟ = 1
⎝ 4⎠
= (1) x 4 + (4) x 3 (21 ) + (6) x 2 (2 2 ) + (4) x(23 ) + (1)( 2 4 )
You do….
Problems for you…..
Page 624
1-9 odds, 17-29 odds
Part 2
Finding Binomial Coefficients in the Calculator
An example….
One more example
You do…
Problems for you….
Page 624 11-16 all (use calculator)
49-67 odds
Problem:
With a simple counting problem, it usually
easiest to list the possible ways the result can
be found.
(3,3), (2,4), (4,2), (1,5), (5,1): 5 ways
distinct means different, no
doubles of the same number
(1,9), (9,1), (2,8), (8,2), (3,7), (7,3), (4,6), (6,4): 8 ways
You try……
For a problem like this it would
be unwise to count all the
possibilities. Use Reason!
(30 possibilities for 1st number) x (30 possibilities for 2nd number) x (30 possibilities for 3rd number)
27,000 possibilities!
How many different local telephone numbers are possible (disregard
the area code)?
Note: The first two numbers of a local number cannot be 0 or 1.
Permutations are the number of ways to order n elements, an
important subset of the Fundamental Counting Principle
You do ….
Remember, with Permutations order matters!
Last problem reworked using the above formula…
Distinguishable permutations mean they are unique
Question:
How many ways can you arrange the letters A,B,C,D?
All permutations are different. 4 x 3 x 2 x 1 = 24 ways
A as 1st letter:
B as 1st letter:
C as 1st letter:
D as 1st letter:
ABCD
BACD
CBAD
DBCA
ABDC
BADC
CBDA
DBAC
ACDB
BCDA
CADB
DCAB
ACBD
BCAD
CABD
DCBA
ADBC
BDAC
CDBA
DABC
ADCB
BDCA
CDAB
DACB
All these arrangements are unique: no repeats
Distinguishable permutations mean they are unique
Question:
How many ways can you arrange the letters A,A,B,D?
AABD where the first A is used and
AABD where the second A is used are NOT
Distinguishable!
AABD BAAD DAAB
ABAD BADA DABA
ABDA BDAA DBAA
AADB ADAB ADBA
12 distinguishable ways or
24 undistinguishable ways
You do ….
Page 634-635
1-27 odds
Part II
Recall that with Permutations, the order of the chosen items was
crucial
When you choose a subset of items from a set, where order
does not matter, then you want Combinations.
10 ways
AB, AC, AD, AE, AF, AG, BC, BD, BE, BF, BG,
CD, CE, CF, CG, DE, DF, DG, EF, EG, FG: 21
ways
Note: this is the same formula as the binomial
coefficient
How many 5-card poker hands are there?
There are 52 cards and you choose 5 (order
does not matter)
You do ….
For both problems order does not matter. You
should start thinking of that first (Does order
matter?) when you look at counting problems.
Choosing seven boys
You do….
Page 634-635
33-55 odds
Let s start by learning the terms.
An experiment is something that happens whose result is uncertain.
The set of all results or outcomes is called the Sample Space of an
experiment.
Any subset of the Sample Space is called an Event.
You do….
The Probability of an event (where all the outcomes are equally likely)
equals the ratio of the outcomes of the event to all the possible
outcomes.
½ because there are two ways for
this to happen (TH or HT) out of a
total of Sample Space of 4 items
(TT,HH,TH,HT), so 2/4.
¼ because there are 13 hearts in a
deck of 52 total cards and 13/52
equals ¼.
Use the Fundamental Counting Principle to Determine the Sample
Space Size
6 x 6 = 36
Figure how many ways to make 7
You do ….
You do….
page 645-646
1- 29 odds
More Probability
Choosing an item at random, means all items are equally likely to
be selected.
Since the graduate is chosen at random, each
graduate is equally likely to be selected.
Number of Colleges in Various Regions
Mutually Exclusive Events have no outcomes in
common.
Rolling a 6 on a die and choosing a spade from a deck of cards are
mutually exclusive.
Rolling a 6 on one die and having the sum of the two dice you rolled equally
4 are not mutually exclusive events.
Mutually exclusive?
No, a card can be both a face card and a heart.
You do….
Mutually exclusive because an employee cannot belong to more than
one row. You cannot have both 0-4 and 5-9 years of service.
You do….
Problems for you…..
Page 646-47
#31-39 odds
Probability – Part 3
If the computer is really generating random numbers, then each
number should be chosen independent of the others.
You do…
The complement of event A, labeled A , is the collection of all
outcomes in the sample space that are NOT in A.
Sometimes it is easier to find the probability of the complement than the
probability of the event you are interested in.
The complement would be that none of the units produced are faulty.
Probability that the unit is produced correctly.
Probability that all 200 are produced correctly.
You do ……
Problems for you……
Pages 647-648
#41-53 odds