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We begin this essay with three tricks. Two of these are due to Art Benjamin.
I got the third one from my new friend Derek Liu of San Diego, CA. As I type
this in January 2013, Derek is six years old. Start with two number written one
below the other. Write their sum just below the second one, and for the fourth,
write the sum of the second and third. Do this until you have written 14 numbers.
Now add the first 10 of these and hide your answer from me. I’ll glance at your
list and instantly tell you your sum. Then compute the ratio of the 10th number
to the 9th number. I can also tell you that to a high degree of accuracy. Now
take the sum of first 12 numbers. I can also tell you that sum without doing the
computation. How? 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 .
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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?
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. Use Binet’s formula to find the number of digits in F2013 .
9. 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.
10. 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 .
11. Let the sequence Gn be defined as above with a = 2. Suppose G15 = 843,
What is G16 ?
∞
∑
Fi
12. Consider the series
. Does it converge? If so, find its sum.
10i
i=1
13. Let Tn denote the number of ways to tile a 2 × n board with dominos. Find
T10 , T11 , and T12 .
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14. Let Zn denote the number of ways to tile a 3 × n board with dominos and one
monomino. Find Z10 , Z11 , and Z12 .
15. 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?
16. 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?
17. How many of the first 2006 Fibonacci numbers are multiples of
(a) 99
(b) 66
(c) 81
18. Find the number of digits in the Fibonacci number Fn .
19. Consider the Zometool triangular arrangement given below.
C
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B
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.
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(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.
20. What proportion of the Fibonacci numbers are even? What fraction of them
are multiples of 3? multiples of 4? 5? Is there a pattern to these answers? I
read this problem at Jim Tanton’s webpage.
21. The first four terms of a sequence are 2, 6, 12, 72. Each terms is obtained from
the previous 2 my multiplying. When the ninth terms is written in the form
2a 3b , what are a and b?
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