Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 579 — #587
14.7. Asymptotic Notation 579
You’ve seen this neat trick for evaluating a geometric sum:
S D 1 C z C z 2 C : : : C z n
zS D z C z 2 C : : : C z n C z nC1
S zS D 1 z nC1
znC1
S D 1
1 z
(where z ¤ 1/
Use the same approach to find a closed-form expression for this sum:
T D 1z C 2z 2 C 3z 3 C : : : C nz n
Problem 14.3.
Sammy the Shark is a financial service provider who offers loans on the following
terms.
Sammy loans a client m dollars in the morning. This puts the client m dollars
in debt to Sammy.
Each evening, Sammy first charges a service fee which increases the client’s
debt by f dollars, and then Sammy charges interest, which multiplies the
debt by a factor of p. For example, Sammy might charge a “modest” ten
cent service fee and 1% interest rate per day, and then f would be 0:1 and p
would be 1:01.
(a) What is the client’s debt at the end of the first day?
(b) What is the client’s debt at the end of the second day?
(c) Write a formula for the client’s debt after d days and find an equivalent closed
form.
(d) If you borrowed $10 from Sammy for a year, how much would you owe him?
Homework Problems
Problem 14.4.
Is a Harvard degree really worth more than an MIT degree? Let us say that a person
with a Harvard degree starts with $40,000 and gets a $20,000 raise every year after
graduation, whereas a person with an MIT degree starts with $30,000, but gets a
20% raise every year. Assume inflation is a fixed 8% every year. That is, $1.08 a
year from now is worth $1.00 today.