Answers to “Cool Problems”
Activity One – Recursive Sequences
A) The first figure has one square, u1 = 1. Each succeeding figure has two
squares added. Recursion: un = un-1 + 2, closed form: un = 2n – 1
B) The first figure has a perimeter of four, u1 = 4. Each succeeding figure adds
four units to the perimeter. Recursion: un = un-1 + 4, closed form: un = 4n
C) The first figure has four segments, u1 = 4. Each succeeding figure adds six segments to the figure. Recursion:
un = un-1 + 6, closed form: un = 6n – 2
Activity Two – Handshake Problem
People
Number of Handshakes
1
0
2
1
3
3
4
6
5
10
Recursion: u2 = 1, un = un-1 + n – 1
closed form: un = (n(n-1))/2
triangular numbers
Activity Three – Credit Card Problem
A) u0 = 1500, which represents the starting account balance (aka seed number)
un = un-1 * 1.015 – 50 will determine the account balance after n months.
B)
Month
Balance ($)
0
1500
1
1472.50
2
1444.59
3
1416.26
4
1387.51
5
1358.32
6
1328.69
C) After 40 months, the balance is $7.64.
D) 40 months x $50/month = $2000. Add the balance remaining ($7.64).
Total of payments = $2007.64
Activity Four – Swimming Pool Problem
A) u0 = 450, which represents amount of the chlorine for the “shock treatment”
un = un-1 * 0.85 + 45 will determine the amount of chlorine after n days.
Day
Chlorine Level (grams)
0
450
1
427.5
2
408.4
3
392.11
10
329.5
20
305.81
40
7.64
41
-42.25
B) The chlorine level will eventually level off after 31 days at 300 grams.
Activity Five - Zookeeper’s Puzzle
A) 15
B) 10
Recursion: u3 = 1, un = n-2 + un-1 closed form: (n-1)(n-2)/2 (triangular numbers)
Activity Six – Tower of Hanoi
A)
Number of Disks
Number of Moves
B)
1
1
recursion: u1 = 1
2
3
3
7
4
15
un = un-1 + 1 + un-1 = 2un-1 + 1
5
31
6
63
7
127
closed form: 2n – 1
Activity Seven – Lots of Dots
A) The step number, n, corresponds to the number of rows. There is one more column than row in each step.
The 20th step will have 20 rows and 21 columns, 420 dots.
B)
recursion: u1 = 2
un = un-1 + 2n
closed form: n(n + 1)
C) Closed (explicit) form: # of Dots = n ( n + 1), given n = step number (# of rows)
Activity 8: The Forester’s Problem
u0 = 4000, which represents the number of trees in the forest
un = Int(un-1 * 0.80 + 1000) will determine the number of trees after n years.
[Note: Int removes fractional parts of trees from calculations]
Year
Number of Trees
0
4000
1
4200
2
4360
3
4488
6
4738
A) About 4738 trees
B) About 4965 trees
C) The forest will eventually stabilize after about 32 years at 5000 trees.
15
4965
Activity 9: The Battle of the Leaves
A)
u0 = 30000, which represents the number of leaves on the front lawn
un = un-1 * 0.70 + 3000 will determine the number of leaves after n days
Day
0
1
2
3
10
15
30
Number of Leaves
30000
24000
19800
16860
10564
10094
10000
B) About 10564 leaves
C) After 30 days, there will be about 10,000 leaves. Lief will not win his battle against Mother Nature.
Activity 10: The Oral Surgeon
A)
u0 = 800, which represents the amount of Trimox (in milligrams) taken on the morning of the procedure
un = un-1 * 0.50 + 200 will determine the amount of Trimox in the body after n days
B) 400 mg
C) After 21 days, change the rule to:
un = un-1 * 0.50 will determine the amount of Trimox in the body after n days
If the patient stops taking the Trimox 21 days after the procedure, then after 30 days there will be less than 1 mg.
in the body.
D) 400 milligrams
E) 400 milligrams
F) There is a “limit,” a maximum amount of Trimox that will remain in the body if the patient takes a daily dose for
a period of time.
Activity 11: Find the Number of Black Triangles (Sierpinski’s Triangle)
Recursion: u1 = 1
un = 3un-1
closed form: un = 3n-1
th
20 triangle contains 1,162,261,467 smaller triangles.