1. CSC2110 Discrete Mathematics Tutorial 4 Number Sequence Hackson Leung

2. Self Introduction You can call me Hackson Email: kmleung@cse.cuhk.edu.hkkmleung@cse.cuhk.edu.hk Office: SHB Room 117 Topics responsible: Number Theory

3. Warm Reminder Homework 1 is released! –Deadline: Oct 19, collect during classes Group project –Group size: 4 –Refer to the course homepage for registration

4. Agenda Summation –Telescoping sum –Arithmetic Series –Geometric Series –Harmonic Series Annuities –Future and Current Values –Return of Annuities

5. Summation Notation All you need to know… If still not familiar, please refer to warm-up tutorial

6. Telescoping Sum To simplify

7. Telescoping Sum By cancelling terms

8. Telescoping sum Example 1: Note:

9. Telescoping sum So,

10. Telescoping sum Example 2: Note: So,

11. Arithmetic Series Given Arithmetic Series Calculate Note that So,

12. Arithmetic Series Calculate

13. Arithmetic Series Calculate

14. Arithmetic Series Calculate

15. Geometric Series Given Geometric Series Don’t use it when r = 1 Infinite Geometric Series for r < 1

16. Geometric Series Calculate

17. Geometric Series Calculate

18. Harmonic Series Definition We say that has no upper bound

19. Future Value I deposit $V in a bank. Interest rate is r%. Bankrate is defined as. After the 1 st year, I will get After the 2 nd year, I will get After the n th year, I will get, which is also known as Future Value

20. Current Value My target is to have $V at the end of the n th year. How much should I deposit today? Current Value:

21. 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 1 st year) Assume it continues forever Current value of the red pocket in the i th year? Total current value?

22. Current Value Example 2 (Attractiveness) 2 plans of investment 1.$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?

23. Current Value Total Current Value of plan 1 Total Current Value of plan 2

24. 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!!

25. 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

26. 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…?

27. 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?

28. Return of Annuities Plan 1: Annual payment For plan 2, money saved is 20,061-12,000 = $8,061

29. Return of Annuities Plan 2 –For investment, you get after 20 years

30. 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!

31. END Thanks!