CSC2110 Discrete MathematicsTutorial 4
Number SequenceHackson Leung
Self Introduction
• You can call me Hackson
• Email: [email protected]
• Office: SHB Room 117
• Topics responsible: Number Theory
Warm Reminder
• Homework 1 is released!– Deadline: Oct 19, collect during classes
• Group project– Group size: 4– Refer to the course homepage for registration
Agenda
• Summation– Telescoping sum– Arithmetic Series– Geometric Series– Harmonic Series
• Annuities– Future and Current Values– Return of Annuities
Summation
• Notation• All you need to know…
• If still not familiar, please refer to warm-up tutorial
Telescoping Sum
• To simplify
Telescoping Sum
• By cancelling terms
Telescoping sum
• Example 1:
• Note:
Telescoping sum
• So,
Telescoping sum
• Example 2:
• Note:
• So,
Arithmetic Series
• Given
• Arithmetic Series
• Calculate
• Note that
• So,
Arithmetic Series
• Calculate
Arithmetic Series
• Calculate
Arithmetic Series
• Calculate
Geometric Series
• Given
• Geometric Series
• Don’t use it when r = 1
• Infinite Geometric Series for r < 1
Geometric Series
• Calculate
Geometric Series
• Calculate
Harmonic Series
• Definition
• We say that has no upper bound
Future Value
• I deposit $V in a bank. Interest rate is r%.
Bankrate is defined as .
• After the 1st year, I will get
• After the 2nd year, I will get
• After the nth year, I will get , which is also known as Future Value
Current Value
• My target is to have $V at the end of the nth year. How much should I deposit today?
• Current Value:
Current Values
• Example 1 (Total Current Value)• Bank rate is 1.05• Each year you receive and deposit $100 red pocket from
your parents (start after 1st year)• Assume it continues forever• Current value of the red pocket in the ith year?
• Total current value?
Current Value
• Example 2 (Attractiveness)
• 2 plans of investment1. $1000 at the beginning of each year
2. $1750 twice a year
• Bank rate is 1.5
• Investment period is 10 years
• Which one is more attractive?
Current Value
• Total Current Value of plan 1
• Total Current Value of plan 2
Return of Annuities
• You borrow $V from a bank, bank rate is b• You want to repay the loan in n years• How much should you pay yearly, at the start of each
year? (Let it be $x)• Idea: Repeatedly subtract x from the loan n times = $0
Total Current Value!!
Return of Annuities
• Example 1– You owe me $109,700– You want to repay it in 15 years– Bank rate is 1.05– Payment is made at the start of each year
• You should pay
Return of Annuities
• Example 1– You owe me $109,700– You want to repay it in 15 years– Bank rate is 1.05– Payment is made at the end of each year
• You should pay…?
Return of Annuities
• Example 2– A car worth $250,000– Bank rate is 1.05– Load period is 20 years– Two plans
1. Borrow $250,000 to buy the car
2. Rent the car for $12,000 annually. Invest money saved to get 5% annual return (rent is paid at the end of each year)
– Which one is better?
Return of Annuities
• Plan 1: Annual payment
• For plan 2, money saved is
20,061-12,000 = $8,061
Return of Annuities
• Plan 2– For investment, you get
after 20 years
Return of Annuities
• Comparison
• Plan 1– If you sell it after 20 years, you can have
$250,000
• Plan 2– For investment, you can get $266,544 after 20
years
• Plan 2 is better!
END
• Thanks!