Sequences

Sequences: An ordered list of numbers that follow a certiain pattern (or rule).

Example:

4, 7, 10, 13, 16, 19, 22

Use the following process to help find a formula for the pattern.

n: is the position or place value
tn: value of that position

When looking at the pattern this way, it is easy to see the it just adding three. So the formula would be.

tn= n-1+3


Or

tn=7n-9



Arithmetic Sequences:

(i) Recursive defintion: An order list of numbers generated by continnous adding value (the common)

(ii) Implictit defintin: An oredered list of numbers where each number n the list is generated by a linear equation.

Common Difference (d):

(i) The number that is repeatedly added to successive term in an arithemic sequence.

(ii) Fommr the implicit definitiion, d is the slope of the linear equation. 

Recursivw: add to what you have. Ex: tn= tn-1+3

Explicit: a formula that uses the position or "n" value to find. Ex: tn= 3n+7

You should also know how to find the common difference

        d = tn - t(n - 1)

        d is the common difference

        tn is an arbitrary term in the sequence

        t(n - 1) is the term immediately before tn in the sequence

As well you should know how to find the nth term in an arithmetic sequence

        tn = a + (n - 1)d

        tn is the nth term

        a is the first term

        n is the "rank" of the nth term in the sequence

        d is the common difference

Perms and Combs, Videos attached

Permutations and combinations are often misused and interchanged comfortably, along with arrangements, collections and groupings. It’s often easy to misuse them, but there is a slight difference in their meanings:

Permutations: An arrangement of a set of events or objects, where in the order of the events MATTERS.

Example: The arrangement of the numbers 1, 2, 3 and 1, 3, 2 is a different PERMUTAION.

Combinations: An arrangement of a set of events or objects, where in the order DOES NOT MATTER.

Example (follow-up): The arrangement of the numbers 1, 2, 3 and 1, 3, 2 is THE SAME COMBINATION.

The knowledge of the difference of meaning in these concepts is crucial.

There are different mathematical formulas for calculating permutation and combination arrangements. It is very important, however, to be able to logically analyze if the order in the arrangement matters.

For permutations:

 For combinations:

The only difference between the two is that in the combination formula, it is essential to divide the result by the number of ways the objects or events can inter-arrange to switch their order, because the order is irrelevant.

The following videos provide a very helpful visual guide to help further develop an understanding of these concepts:

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-1

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-2

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-3

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-4

perms and combs

Permutations, has several different meanings, but all are related to “the act of permuting”, rearranging objects and values. They occur in almost every domain of mathematics. It is an arrangement of numbers in an order . For example there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1).

The formula for permutation is:

                                nPr =n!/(n-r)

Combination- is a way of selecting several things out of a large group where the order doesn’t matter. In smaller cases you can count the number of combination. When the set gets larger you have to use difficult mathematics to find the number of combinations.

The combination formula:

              (nk)=n(n-1)…(n-k+1)/k(k-1)…1

Premutations and Combinations

Permutations- A permutation is a order of numbers where the order does matter
Combinations- A Combination is a order of numbers where the order does not matter

A combination lock is a perfect example of a Permutation as it is three order number line where the numbers have to be in the right order to open the lock.

If your combo is 23-67-9 you can only put it in as this order if you put 67-23-9 the numbers are the same but the order is not making it impossible to open your locker.

The order through which you put fruit into a smoothie is a combination
If Apples are 1
Banannas are 2
Peaches are 3
The order can go many ways
123
132
321
213
231
312

In other words a permutation is an ordered combination

The above formula is for Permutations without repitition the below fomule is for combinations without repitition.

where n is the number of things to choose from, and you choose r of them       (Order does not matter)

For things where repitition is allowed the formulas are as follows

Permutations is simply N to the exponent R where N is the number of things to choose from and R is how many you will choose

Combinations where repitition is allowed is shown as the formula below

where n is the number of things to choose from, and you choose r of them
(Repetition allowed, order doesn't matter)

All information was taken from the link below the above descriptions and formulas are a short sumamry of the site the site as well provides real world examples

http://www.mathsisfun.com/combinatorics/combinations-permutations.html

Combinations & Permutations.

Combinations are easy going. Order doesn’t matter. You can mix it up and it looks the same. Let’s say I’m a cheapskate and can’t afford separate Gold, Silver and Bronze medals. In fact, I can only afford empty tin cans.
How many ways can I give 3 tin cans to 8 people?
Well, in this case, the order we pick people doesn’t matter. If I give a can to Alice, Bob and then Charlie, it’s the same as giving to Charlie, Alice and then Bob. Either way, they’re going to be equally disappointed.
This raises an interesting point — we’ve got some redundancies here. Alice Bob Charlie = Charlie Bob Alice. For a moment, let’s just figure out how many ways we can rearrange 3 people.
Well, we have 3 choices for the first person, 2 for the second, and only 1 for the last. So we have 3 * 2 * 1 ways to re-arrange 3 people.
Wait a minute… this is looking a bit like a permutation! You tricked me!
Indeed I did. If you have N people and you want to know how many arrangements there are for all of them, it’s just N factorial or N!
So, if we have 3 tin cans to give away, there are 3! or 6 variations for every choice we pick. If we want to figure out how many combinations we have, we just create all the permutations and divide by all the redundancies. In our case, we get 336 permutations (from above), and we divide by the 6 redundancies for each permutation and get 336/6 = 56.
The general formula is

which means “Find all the ways to pick k people from n, and divide by the k! variants”. Writing this out, we get our combination formula, or the number of ways to combine k items from a set of n:

http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

Enjoy!

Permutations

Permutations - An arrangement of 'n' (number) objects taking all or some of the objects at a time where the order does matter.

Without Repetition:

            Formula:  nPr = n!/(n-r)!

n - The total number of items to choose from.
r - The number of items being chosen.

ex. a) How many ways can you order billiard balls numbered 1 to 16?
 b) How many ways can you order 5 different balls?


a)
An easy way to do it is insert [1],[6],[MATH], go over to the 'Probability' option and select option 4 so it will look like this on your calculator:

16!

Really all '!' is, is a short cut for going 16x15x14x13x12xx11x10x9x8x7x6x5x4x3x2x1
answer: 20922789888000 or 2.09x10¹³

b)
 You can use the formula nPr = n!/(n-r)! 
n=16
r=5                     =16!/(16-5)!
                           =16!/11!
                           =524160
or with your calculator you can take a shortcut by inserting "16" then [MATH] button, over to probability and select option 2 "nPr"than insert the number "5". It should look like this on your calculator.

16 nPr 5

Hit solve and you should get the same answer as above: "524160".

BUYING AND LEASING A CAR TERMS


BUYING:TO EXCHANGE MONEY FOR ITS EQUIVALENT


LEASING: A CONTRACT AGREEMENT TO USE A CAR OVER A PERIOD OF TIME , AND MAKING PAYMENT WHILE USING THE CAR.




        BUY AND LEASING A CAR


    PRO'S OF BUYING A CAR:


   IN THE END YOU OWN THE CAR
   YOU CAN DO AS MANY MODIFICATIONS AS YOU WANT
    YOU CAN PUT AS MY KILOMETERS ON THE CAR AS YOU LIKE


     
      CON'S OF BUYING A CAR:
ITS EXPENSIVE
YOU HAVE TO PAY FOR ALL THE REPAIRS

    PRO'S TO LEASING A CAR:
NOT AS EXPENSIVE AS BUYING A CAR
YOU DON'T HAVE TO PAY FOR REPAIRS ON THE CAR
YOU GET A NEW CAR EVERY 3 YEARS    CON'S TO LEASING A CAR:
YOU DON'T OWN THE CAR IN THE END
YOU CAN'T MODIFY THE CAR
YOU HAVE LIMITED KILOMETERS YOU CAN PUT ON THE CAR
YOU CAN'T HAVE ANY DENTS OR SCRAPES IN THE CAR WHEN YOU BRING IT BACK TO THE DEALER]

Probability – The chance or likelihood that something will happen.

Probabilities must be between 0 and 1

·         A probability of 0 means it’s impossible

·         A probability of 1 means it will definitely happen

Probabilities can be expressed as…

·         Ratio

·         Fraction

·         Decimal

·         Percent

Outcome – The result of an experiment

Sample Space – All the possible outcomes of an experiment

Event – One or more outcomes of an experiment

Type of Events

1.      Independent Event – Events that is not affected by any other events.

Ex.  A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die?

        P(head) = ½

        P(3) = 1/6

        P(head and 3) = P(head) x P(3)

                                    = (1/2) X (1/6)

                                 = 1/12

2.      Dependent Event – Events that can be affected by previous events.

Ex. A chosen is chosen at random from a standard deck of 52 cards. Without replacing it, a second card is chosen. What is the probability that the first chosen card chosen is a queen and the second card is a jack?

       P(queen on first pick) = 4/52

       P(jack on 2^nd pick given queen on 1^st pick) = 4/51

       P(queen and jack) = (4/52) x (4/51)

                                      = 16/2652

                                      = 4/633

3.      Simple Event – A single outcome in the sample space

Ex. What is the probability of rolling a 5?

        P(5) = 1/6

4.      Compound Events – More than one outcome in the sample space

Ex. What is the probability of rolling an even number?

       Ex. P(even) = 3/6

                           = 1/2

Factorial – The result of multiplying a series of decreasing numbers

Note: The factorial for 0 is always 1

Ex. 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362 880

Ex. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

Probability

Probability = (desired outcome)                   Ex.   P(heads)= 1/2 = 50% = .5
                   ( total # of outcomes)


The types of probability:
1. Theoretical- what the results should look like in theory.
 Ex. flipping a coin and getting heads is 1/2 or 50%


2. Experimental- The results are found out in tests.
Ex. flipping a coin 100 times and getting heads 56 times


This type of probability isn't accurate if very few trials are done.


Tree diagrams:
The outcomes for 2 probabilities; flipping a coin and rolling a die.


or it can be written in a table called a sample space


Sample space
H-1       T-1
H-2       T-2
H-3       T-3
H-4       T-4
H-5       T-5
H-6       H-6

To find the probability of 2 events occurring at the same time you multiply both the probabilities together.


Ex. Probability of getting heads is 1/2 and rolling a 5 is 1/6 so 1/2 x 1/6 = 1/12 

Using TVM solver

N: Total # of payments to the account.
I%: Annual interest rate as a '%'.
PV: Present value of the account.
PMT: Payments made to the account.
FV: Future value of the account.
P/Y: # of payments made per year.
C/Y:  # of compounded periods per year.
PMT: Depends when payments are made.

Net Worth

Net Worth ( synonymous with equity ) - the difference between the value assets(what you own) and liabilities(what you owe).

Net worth can be a useful tool to measure your financial progress from year to year.

Three categories of Assets:

a) Liquid assets (sometimes called Current Assets) - assets that can be converted into cash quickly and without financial penalty. Cash accounts, treasury bills, money market funds, Canada Savings Bonds are all invested vehicles found in this category.

b) Semi-liquid Asset - include longer term investments that are intended to sore up value for major future needs such  as education costs or retirement. It may take a while to convert it to cash and may pay fee. Some examples are: stocks, bonds, mutual funds, real estate ( other than your principal residence ), RRSPs and registered pension plans (RPPs).

c) Non-Liquid Assets - are items you acquire for your family's long term use or enjoyment. May include you home, vacation property, cars bouts, antiques, and furnishinhs.

Two types of Liabilities:

a) Short-term Debt - are all debts that must be paid within the next twelve months. Credit card balances, personal, installment and consumer loans fall into this category.

b) Long term Debt - are used for two purposes: to finance long-term investments such as real estate or the purchase of a major personal assets like you residence, vacation property or long term car loans.

Debt-equity ratio:

-Debt includes all debt ( short term or long term ) except the mortgage on you principal residence.

-DER should not exceed 50% of net worth.

                           Debt Equity Ratio (DER) = Liabilities - Mortgage

                                                                            Net Worth

more info:

http://financialplan.about.com/od/personalfinance/ht/networthhowto.htm

http://www.msmoney.com/mm/financial_health/where_stand/yournetworth.htm

Buying a House or Rent

Advantage of Buying a House

-You end up being the owner

-Building Equity, Money will be yours

-Able to renovate and change as you choose

-Pets are your choice

-Homes usually increase in value over time (more equity/money)

-Parties and friends on your schedule

Disadvantage of Buying a House

-All responsibility is on you (Repairs, Money, Bills..)

-You might go into debt

-Need a down payment

-Bad Neighbours

-Interest on mortgage is more than renting

-"Other" Expenses (Insurance, Security…)

Advantage of Renting

-Cheaper

-No Maintenance

-No Down Payment

-Short Stay

- No tax

Disadvantage of Renting

-Bad Landlord

-Some don't allow pets

-No Changes

-Bad Neighbours

-Roommate

-Have to live by 'their' rules

-One Year Lease

-All your money is Lost

More Info:

http://warriorism.info/personal-finance-advice/renting-vs-buying-a-home.php

http://www.haesemathematics.com.au/samples/ibssl-2_15.pdf

Real-Life Examples:

http://www.getrichslowly.org/blog/2007/07/16/renting-vs-buying-the-realities-of-home-buying/

http://www.financefox.ca/owning-home-renting/

Example:

Determine the maximum affordable price a family should pay to purchase a home if they have a gross monthly income of $3600.00; monthly property taxes are $150.00, and monthly heating costs average $135.00 per month all year. The family is able to finance their mortgage at 7% and has $15000.00 available to put into a down payment.

Interest Rate Factor Table*

Rate Factor Rate Factor

6.0% 0.00640 8.0% 0.00763

6.5% 0.00670 8.5% 0.00795

7.0% 0.00700 9.0% 0.00828

7.5% 0.00732 9.5% 0.00861

Source:

http://realestate.yahoo.com/info/guides/buying-vs-renting

Buying vs Leasing

Trying to decide if buying a car or leasing a car can be difficult but educating yourself a little can help you out in the long run.

 Advantages to Buying a car

• You own the car
• You can make any modifications you want
• You can drive it as many kilometres as you please
• You can drive the vehicle for as many years as you want
• Buying is cheaper in the long run if you drive it for many years
• Buying is also cheaper if you plan to buy it outright without taking a loan

Advantages to Leasing a car

• Lower initial costs 
• Lower monthly payments through leasing a car versus financing a loan
• Repairs (that are not your fault) are the responsibility of the owner
• You like to drive a new and fairly trouble free vehicle
• No resale or trade in obligation

Disadvantages to Buying a car

• Financing is very expensive versus the low monthly lease payments.
• New cars are VERY expensive may have to settle for an old model
• Bound to a fix termed finance plan
• Down Payment can be costly

Disadvantages of Leasing a car

• More expensive in the long run if you choose to buy the car for residual value after the lease is up. You end up paying more than the car is valued at initially.
• No modifications Minor or Major
• Set amount of kilometres

With these pros and cons in mind deciding which one to choose is a good suggestion. If you have to have the best car on your block then leasing may be for you. But if you want to drive it for a long time and own it then buying a car will be the right path. Below is an examples of payment plans.

Leasing: A new luxury car has a purchase price of $24,500 and can be leased for 36 months with no down payment for $345 a month plus taxes. After the 36 months the car has a purchase option price of $14,500.

$345 (monthly payments) x 1.12 (tax) = $386.40 ( monthly payment + tax) x 36 months = $13,910.40 at end of the lease.
If you want to purchase the car it would now be $13,910.40+$14,500= $28,410.40 in total by the time the car is yours.

Buying: You can take out a loan to buy the car outright at an interest rate of 1.8% compounded monthly. When you buy the car you pay GST, PST and a down payment of $3000. Using the TVM solver we can discover monthly payments then the total cost of purchasing the car.
N=36 (Number of payments made to the account)
I%=1.8 (Interest rate)
PV=$27,440 (this is the initial car cost of $24,500 x 1.12 for tax) or the loan itself
PMT=$783.56 (monthly payment)
FV=0 (Future value of the loan)
P/Y=12 (Number of payments per year)
C/Y=12 (Number of compound periods per year)
Beginning or End


Now that we know monthly payments we can see how much we payed after the three years.
36 months x $783.56 (monthly payment) = $28,208 + $3000 (down payment) = $31,208 total after the three years.

To calculate how much was interest we simply subtract $27,440 (loan) from the total payed at the end of the three years. $28,208-$27,440= $786 dollars payed in interest

By the end in this situation that leasing the car is cheaper you own it after the lease is up and the only difference is the $3000 down payment you used when buying the car so it would be smarter to lease this car instead of buying it outright.

 A great site to find out how to build a car you enjoy and see what's in your budget is http://www.ford.ca/app/fo/index.do  this site  will allow you to build your own Ford vehicle and then choose payment options it gives you the option of financing and leasing to see which one is better for you.

Personal Finance

• Buying a House

Terms:
     - Principal- how much you borrowed from a bank (found by subtracting, the down payment form the selling price)
      -Down payment- your money that you use to help pay for the house. This is becomes your equity.
      -Mortgage- a loan secured by property
      -Equity-share of the value of the house
               Example: if you owe $100,000 on a $300,000 home, you have $200,000 of equity on the day you move into the house your down payment is your equity.
      -Interest- the amount you pay the bank to borrow their money. It also called "the cost of borrowing"
      -Mortgage payment- a regular payment to the bank that pays for the interest and pays down the unpaid balance. (monthly, semi-monthly, biweekly, weekly)
      -Amortization period- how long will it take to pay off the mortgage, often 25 years
      -Term- the length of time a mortgage agreement is good for. Anywhere from 6 months to 10 years. When the term is over, a new agreement must be reached with a new interest rate.




 Initial Cost of Buying a House:
- Appraisal fees - When borrowing money the lender (e.g., bank) must determine the value of the property. A certified appraiser will determine the value of the property.
- Inspection costs- An inspection of the property is not absolutely necessary, but it will let you know if any repairs are required or if the house has any structural problems.
- Property survey- This will supply information on how buildings, fences, and the like are situated on the property. If there are any easements on your property, it is a good idea to know about this before making the purchase. Easements are rights of way by the town, city, or utility company to access your land for specific purposes such as digging up telephone wires. Anencroachment is an intrusion onto your land by a neighbour's structure, or possibly an encroachment on your neighbour's land by something on your property. In either case, you would certainly want to know about this before purchasing this property. If a recent survey is available to you, a property survey may not be necessary.
- Insurance costs for high ratio mortgages - You must pay additional insurance costs if you have a high ratio mortgage. A high ratio mortgage is a house loan where less than 25% of the original cost of the home is paid with the down payment. The cost for this insurance is usually about 1.25% -3% of the total mortgage, depending upon the amount of your down payment.
- Home insurance - As soon as you purchase a home, it is wise to purchase home insurance.
- Land transfer tax- Some provinces levy a tax on any property that changes hands. As the buyer, you are responsible for this cost. It is usually a small percentage of the purchase price.
       0-$30,000         nothing
  $30,000-$90,000    0.5%
  $90,000-$150,000  1.0%
  over $150,000        1.5%
- Interest adjustments- The buyer is responsible for any interest payable between the closing date (the date of possession) and the first mortgage payment.
- Prepaid property taxes and utilities- You will have to reimburse the seller for any utilities or taxes paid for the period of time you own the home.
      month/12*annual amount
- Legal fees - It is normally a good idea to hire a lawyer you trust to look after all legal transactions.
- Sales tax- GST may be charged when buying a new home in Manitoba.
- Moving expenses - You may need to pay professional movers, rent a truck, or hire helpers when you move. Driving expenses, meals, and motel bills may also be part of the cost of moving.
- Service charges- Hookup fees for telephone, TV, and utilities will likely be added to your first bills.
- Immediate repairs- Some of these may be necessary prior to your moving in. You may want to negotiate the cost of these repairs with the seller.
- Appliances - You may need to buy appliances such as a fridge, stove, washer, dryer, and/or dishwasher when you move in.
- Decorating cost- You may want to do some painting, wallpapering, carpeting,before you come in.

• Gross Debt Service Ratio (GDSR)

Your GDSR should not exceed 32%-highest amount you should spend
GDSR= (monthly mortgage+ monthly taxes+ monthly heat)/gross pay*100%

•  TVM Solver

    N=total number of payment to the account
    I%=annual interest rate as a percent
    PV=present value of the account
    PMT=payments made to the account
    FV=future value of the account
    P/Y=number of payments made per year
    C/Y=number of compounding periods per year

• Buying vs Leasing a New Car

1. Buying a car

When calculating the price of a new car, we first have to subtract the value of any trade-ins. Then we add the GST (5%) and PST (7%) to the difference to find out how much we have to pay.

  Example：Steve wants to buy a new Ford truck worth $22,000 and he is offered $3000 for a trade-in value on his old truck.How much does he pay?

     $22,000-$3000=$19,000

     $19,000*(1+0.05+0.07)=$21,280

2. Leasing a car

 When you lease a car you are only paying for the depreciation of the car during the lease. This means you are paying for the lost value of the car. You also must pay the PST and GST on the depreciation and include the taxed in your monthly payment.

 At the end of the lease, the value of the car is now called the residual value. The residual value is a percentage of the original price.

  

• Net Worth

Net worth- defined as the difference between your total assets and liabilities. 

- Assets- what you own, include cash, bank accounts, stocks, mutual funds...

- Liabilities- what you owe. It includes mortgage, car loan, personal loans, line of credit...

3 types of Assets

- Liquid Asset- anything that can be converted to cash quickly.

                        For example, cash, Canada saving bonds...

- Semi-liquid Asset- longer term investments meant for the future

                        For example, bonds, stocks, real estate, mutual funds...

- Non-liquid Asset- investments for long term and enjoyment and include things like home, vacation property, cars, boats...

2 types of Liabilities

- Short term debt- anything to be paid off in the next 12 months

                           -include credit card, personal loans, consumer loan

- Long term debt- anything that takes a long time to pay off

                          - car loan, mortgage

Debt Equity Ratio (DER)- this is a measure of debt burden. You should try to keep your DER under 50%.

DER=(total liabilities - mortgage)/ net worth*100%

Net worth= total assets- total liabilities

We want to improve DER by keeping our liabilities small and increasing our net worth.