CSC2110 Discrete MathematicsTutorial 4

Number SequenceHackson Leung

Self Introduction

• You can call me Hackson

• Email: kmleung@cse.cuhk.edu.hk

• 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!