Tutorial 11 of CSCI2110
Number of Sequence & Recursion
Tutor: Zhou Hong (周宏)
hzhou@cse.cuhk.edu.hk
Announcement
• HW2 & HW3 has been marked, please pick them up.
• HW4 will be returned in next Monday.
Announcement
• In your homework, you should declare your
cooperators or cite references, this will not affect
your grades.
– Note: write your OWN solutions, do NOT just copy!
• Fail to do so may be considered as plagiarism.
– The same notation or the same term (e.g. “imaginary women”)
are clues.
Outline
• Number of Sequence
• Setup Recurrence Relations
NUMBER OF SEQUENCE
Warm Up Exercise
What’s the next number?
• a1 = -3, a2 = -1, a3 = 1, a4 = 3, …
Answer: a5 = 5
• a1 = 3, a2 = 6, a3 = 12, a4 = 24, …
Answer: a5 = 48
What about this one?
• a1=1, a2=11, a3=21, a4=1211, a5=111221, a6=312211, a7=13112221, …
Answer: a8=1113213211
• This is just for fun = )
Warm Up Exercise (Cont.)
Write down the general formula of ai
• a1 = -3, a2 = -1, a3 = 1, a4 = 3, …
Answer: ai = a1 + (i-1)d = -3 + 2(i-1) = 2i - 5
• a1 = 3, a2 = 6, a3 = 12, a4 = 24, …
Answer: ai = a1 q(i-1) = 3 x 2(i-1)
Write down the closed form of the following partial sum (
𝒏
𝒊=𝟏 𝒂𝒊 )
• a1 = -3, a2 = -1, a3 = 1, a4 = 3, …, ai = 2i – 5, …
Arithmetic Sequence:
𝒏
𝒊=𝟏 𝒂𝒊
=
𝒂𝟏 +𝒂𝒏 𝒏
𝟐
=
𝟐𝒂𝟏 + 𝒏−𝟏 𝒅 𝒏
𝟐
• a1 = 3, a2 = 6, a3 = 12, a4 = 24, …, ai = 3 x 2(i-1), …
Geometric Sequence:
𝒏
𝒊=𝟏 𝒂𝒊
=
𝒂𝟏 (𝟏−𝒒𝒏 )
𝟏−𝒒
= 𝟑 × 𝟐𝒏 − 𝟑
= 𝒏𝟐 − 𝟒𝒏
Warm Up Exercise (Cont.)
Evaluate the following summation (Hint: Telescoping Sum)
•
𝑛
1
2 +8𝑘+3
4𝑘
𝑘=0
Answer:
=
1
=
2
1
=
2
𝑛
1
𝑘=0 (2𝑘 + 1)(2𝑘 + 3)
𝑛
1
1
−
2𝑘 + 3
𝑘=0 2𝑘 + 1
1
𝑛+1
1−
=
2𝑛 + 3
2𝑛 + 3
Future Value
Given bank rate b unchanged, if we deposit $X now, how much will
we have after n years?
Now
After 1 Year
After 2 Years
…
After n Years
$X
$X * b
$X * b2
…
$X * bn
Future Value:
1 dollar today will be worth bn dollars after n years.
Current Value
Given bank rate b unchanged, if we want to have $X after n
years, how much should we deposit now?
Now
…
After n-2 Years
After n-1 Years
After n Years
$X / bn
…
$X / b2
$X / b
$X
Current Value:
1 dollar after n years is only worth 1 / bn dollars today.
Mortgage Loan
Suppose the bank rate is b and keeps unchanged. Now, we rise a
mortgage loan of $X dollars to buy a house. We have to repay the
loan in n years. What is the annual repayment $p. (assume
repayment is made at the end of each year)
Solution 1: Consider Future Value
Year
Amount to Be Repaid
Now
$X
1 year
$X * b - $p
2 years
($X * b – $p) * b – $p = $X * b2 – $p * b – $p
…
…
n years
$X * bn – $p* bn-1 – … – $p * b – $p
In order to repay the loan in n years, we must have
$𝑋 ⋅ 𝑏 𝑛 − $𝑝
𝑛−1
𝑖=0
𝑏𝑖 = 0
⇔
𝑛
1
−
𝑏
$𝑋 ⋅ 𝑏 𝑛 − $𝑝
=0
1−𝑏
Therefore, in each year, we have to repay
𝑏 𝑛 (1 − 𝑏)
$𝑝 = $𝑋
1 − 𝑏𝑛
Solution 2: Consider Current Value
Year
Current Value of Each Year’s Payment
1 year
$p / b
2 years
$p / b2
…
…
n years
$p / bn
The key observation is that the total current value should equal to $X.
Let 𝑟 =
1
,
𝑏
$𝑋 =
𝑛
𝑟(1
−
𝑟
)
𝑖
$𝑝 ⋅ 𝑟 = $𝑝
1−𝑟
𝑖=1
𝑛
Therefore, in each year, we have to repay
1−𝑟
𝑏 𝑛 (1 − 𝑏)
$𝑝 = $𝑋
= $𝑋
𝑛
𝑟(1 − 𝑟 )
1 − 𝑏𝑛
RECURSION
Disclaimer: Some of the slides in this section are taken from the slides by Chow Chi Wang,
last year’s Tutor of CSCI2110.
Recursion
• Recursion is an important technique in computer
science. The key idea is to reduce a problem into the
similar problems in simpler cases.
• In this tutorial, we will focus on how to setup the
recurrence relations. We will discuss solving
recurrence relations in next week’s tutorial.
• Tip: After setting up a recurrence relation,
remember to test it’s correctness by trying some
base cases.
Fibonacci Variant
We have a single pair of rabbits (male and female)
initially. Assume that:
– (a) the rabbit pairs are not fertile during their first month
of life, but thereafter give birth to four new male/female
pairs at the end of every month;
– (b) the rabbits will never die.
Let 𝑟𝑛 be the number of pairs of rabbits alive in the 𝑛-th month.
Find a recurrence relation for 𝑟0 , 𝑟1 , 𝑟2 …
Fibonacci Variant
Initiation
(𝑟0 = 1)
Month 1
(𝑟1 = 1)
Month 2
(𝑟2 = 5)
(𝑟3 = 9)
Month 3
Month 4
(𝑟4 = 29)
Key:
Baby rabbit pair
Fertile rabbit pair
Fibonacci Variant
• Let 𝑏𝑛 be the number of baby rabbit pairs in the 𝑛-th
month.
• Let 𝑓𝑛 be the number of fertile rabbit pairs in the 𝑛-th
month.
Note that we have:
where,
𝑟𝑛 = 𝑓𝑛 + 𝑏𝑛 ,
𝑓𝑛 = 𝑓𝑛−1 + 𝑏𝑛−1 = 𝑟𝑛−1 ,
𝑏𝑛 = 4 ∗ 𝑓𝑛−1 = 4 ∗ 𝑟𝑛−2
Therefore:
𝑟𝑛 = 𝑟𝑛−1 + 4 ∗ 𝑟𝑛−2 for 𝑛 ≥ 2, and 𝑟0 = 𝑟1 = 1.
The Josephus Problem
Flavius Josephus is a Jewish historian living in the 1st century.
During the Jewish-Roman war, he and his 40 soldiers were trapped in a
cave, the exit of which was blocked by Romans.
The soldiers chose suicide over capture and decided that they would
form a circle and proceeding around it, to kill every third remaining
person until no one was left.
However, Josephus wanted none of this suicide nonsense.
His task is to choose a place in the initial circle so that he is the last
one remaining and so survive.
The Josephus Problem in General Form
There are n people numbered 1 to n around a circle. Start with people #1,
eliminate every j-th remaining people until only one survives. The task is to
find the survivor’s initial number.
A simple case with n=10 and j=2:
The elimination order: 2, 4, 6, 8, 10, 3, 7, 1, 9.
So, people #5 is the survivor.
The Case of j=2
Let J(n) be the survivor’s number when start with n people.
If n is a even number, we have
J(2k) = 2J(k) – 1, for k >= 1.
If n is a odd number, we have
J(2k+1) = 2J(k) + 1, for k >= 1.
Base case, J(1) = 1.
The General Value of j
Consider eliminating people one by one. In the very beginning, the j-th
people is eliminated, then we start the next count with the j+1-th
people starting with people #1.
J(n-1) is the survivor’s number in the remaining circle of n-1 people,
starting with the j+1-th people in the initial circle.
Then, the survivor is the (J(n-1)+j)-th people starting with people #1
in the initial circle.
Note that, the position of the i-th people in the initial circle is
((i-1) mod n) + 1 , starting with people #1.
Therefore, we have the following recurrence relation
J(1) = 1,
J(n) = ((J(n-1)+j-1) mod n) + 1, for n >= 2.
Comparison of The Two Recurrence Relations
J(1) = 1,
J(n) = ((J(n-1)+j-1) mod n) + 1, for n >= 2.
V. S.
J(1) = 1,
J(2k) = 2J(k) – 1, for k >= 1,
J(2k+1) = 2J(k) + 1, for k >= 1.
Thank You!
Q&A?