Download Exploring Fibonacci Numbers

Exploring Fibonacci Numbers
Awesome Math, 2011
This problem set is designed as an introduction to the famous and ubiquitous sequence of integers called the Fibonacci sequence. The sequence is
defined by letting F1 = 1 and F2 = 1, and the recurrence relation Fn =
Fn−1 + Fn−2 for all n ≥ 3. Thus, F3 = 2, F4 = 3, F5 = 5. The first few
terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, F12 = 144. For convenience, we build a table of the first 20 values below:
1
1
2
3
5
8
13
21
34
55
89 144 233 377 610
987 1597 2584 4181 6765
Our problems are essentially of two types, divisor problems and addition
problems. Each problem requires both experimentation and proof. The
proofs typically use the Principle of Mathematical Induction (PMI). PMI
says that if a proposition P about the nonnegative integers is true about 0
(or in some cases 1) and it is also true about the successor of each integers
for which it is true, then it is true about all the positive integers. We’ll
provide a few sample proofs as we go. Incidentally, there is another common
sequence of numbers defined similarly, called the Lucas Numbers, Ln about
which we can compose similar problems. First L1 = 1 and L2 = 3. Also,
Ln = Ln−1 + Ln−2 for all n ≥ 3.
1. A monomino is a unit square (¥), and a domino (¥¥) is a pair of
monominoes glued together along a common edge. How many ways is
there to fill up a 1 × 7 rectangle with monominoes and dominoes?
2. Experiment with the Fibonacci sequence to find a formula for the sum
Sk of the first k Fibonacci numbers. Symbolically,
Sk = F1 + F2 + · · · + Fk .
This sequence begins S1 = 1, S2 = 2, S3 = 4, S4 = 7, S5 = 12. Find a
pattern.
3. Find the units digit of the Fibonacci number F2008 .
4. How many of the first Fibonacci numbers F1 , F2 , . . . , F1000 are multiples
of 9?
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Exploring Fibonacci Numbers
Awesome Math, 2011
5. What is the sum of the first 1000 even Fibonacci numbers? For example, the sum of the first 2 Fibonacci numbers is F3 + F6 = 2 + 8 = 10.
You may express your answer in terms of other Fibonacci numbers.
6. What is the sum of the first 100 evenly subscripted Fibonacci numbers?
For example, the sum of the first 4 evenly subscripted Fibonacci numbers is F2 + F4 + F6 + F8 = 1 + 3 + 8 + 21 = 33. You may express your
answer in terms of other Fibonacci numbers.
7. Binet’s formula. Let α =
√
1+ 5
2
and β =
√
1− 5
.
2
Then
√
Fn = (αn − β n )/ 5
for all n ≥ 1. Use this formula to verify that F10 = 55 and F20 = 6765.
8. The first Fibonacci number that is divisible by 11 is F10 = 55. Find
the second and the third. Develop a conjecture about which Fibonacci
numbers are multiples of 11 and prove it.
9. In this problem, and in the one that follows, we start with G1 = a
and G2 = b, where a and b are positive integers. We define a sequence
just as above by the recurrence Gn+2 = Gn + Gn+1 . Now define yet a
third sequence Hn as follows: Hn = Gn + Fn , where Fn denotes the nth
Fibonacci number. Prove that the new sequence Hn also satisfies the
recurrence relation Hn+2 = Hn + Hn+1 .
10. Let the sequence Gn be defined as above with a = 2. Suppose G15 =
843, What is G16 ?
11. Consider the series
∞
X
Fi
. Does it converge? If so, find its sum.
i
10
i=1
12. Let Tn denote the number of ways to tile a 2 × n board with dominos.
Find T10 , T11 , and T12 .
13. Let Zn denote the number of ways to tile a 3 × n board with dominos
and one monomino. Find Z10 , Z11 , and Z12 .
14. Let S be a subset of the set {1, 2, 3, 4, . . . , 100}, and suppose that no
three elements of S are the lengths of the sides of a non-degenerate
triangle. What is the largest number of elements S can have?
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Exploring Fibonacci Numbers
Awesome Math, 2011
15. Let K be the integer you found in the problem above. How many
subsets of {1, 2, 3, 4, . . . , 100} with K elements satisfy the property that
no three of its members are the lengths of the sides of a non-degenerate
triangle?
16. How many of the first 2006 Fibonacci numbers are multiples of
(a) 99
(b) 66
(c) 81
17. Find the number of digits in the Fibonacci number Fn .
18. Consider the Zometool triangular arrangement given below.
C
..•..
... .....
.. ...
... ....
... y
..
...
...
...
..
...
z ..
...
..
...
..
.•E
.
.
.
.
.
..
x . ....
.. D ............ ....
.
.•.
..
...... .............1 .... 1
... .1...........
....... ...
....... ..
.. ........
.........
.•.......
.
•
A
B
y
The structure is built with blue struts of lengths x, 1, y, z, where x <
1 < y < z. Triangles ABD and AEC are similar, and triangles ABC,
DEC, and BEA are similar.
(a) Find the lengths of y and z in terms of x.
(b) Solve for x, y and z.
(c) Solve x2 − x − 1 = 0 for two roots. Call the smaller on β and the
larger one α. Compute each of the numbers α2 , α3 , α4 , α5 , writing
each of these in terms of α itself. Then do the same for β.
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