- 1. What is the definition of each element in the Fibonacci sequence?
A) The average of the two elements that precede it
B) The product of the two elements that precede it
C) The difference between the two elements that precede it
D) The sum of the two elements that precede it
- 2. How are Fibonacci numbers commonly denoted?
A) Fi
B) Fn
C) Nf
D) Fb
- 3. What are the standard initial elements F1 and F2 of the Fibonacci sequence?
A) 0 and 1
B) 1 and 1
C) 1 and 2
D) 0 and 0
- 4. Which number is the sixth element in the sequence starting with F0 (0, 1, 1, 2, 3, 5, ...)?
A) 13
B) 5
C) 3
D) 8
- 5. In which ancient mathematics were Fibonacci numbers first described?
A) Indian mathematics
B) Roman mathematics
C) Chinese mathematics
D) Greek mathematics
- 6. Who first described the Fibonacci numbers around 200 BC in connection with Sanskrit poetry?
A) Pingala
B) Fibonacci
C) Jacques Philippe Marie Binet
D) Édouard Lucas
- 7. What was the context of the earliest description of Fibonacci numbers in Indian mathematics?
A) Sanskrit poetry patterns
B) Rabbit population growth
C) Planetary orbits
D) Financial interest rates
- 8. Who introduced the Fibonacci sequence to Western European mathematics?
A) Leonardo of Pisa
B) Johannes Kepler
C) Daniel Bernoulli
D) Abraham de Moivre
- 9. In which book did Fibonacci introduce the sequence to Western Europe?
A) Fibonacci Quarterly
B) OEIS A000045
C) Liber Abaci
D) Natya Shastra
- 10. What is the name of the academic journal entirely dedicated to the study of Fibonacci numbers?
A) The Golden Ratio Review
B) Fibonacci Quarterly
C) Liber Abaci Journal
D) Sequence Mathematics
- 11. Which of the following is an application of Fibonacci numbers in computer science?
A) Fibonacci heap data structure
B) Hash table collision resolution
C) Binary search tree
D) Dijkstra shortest path algorithm
- 12. What are the graphs called that use Fibonacci numbers for interconnecting parallel and distributed systems?
A) Golden ratio networks
B) Fibonacci trees
C) Binet graphs
D) Fibonacci cubes
- 13. Which biological setting does NOT mention the appearance of Fibonacci numbers?
A) Arrangement of leaves on a stem
B) Flowering of an artichoke
C) Fruit sprouts of a pineapple
D) Root structures of ferns
- 14. What constant is strongly related to Fibonacci numbers, appearing in Binet's formula?
A) Golden ratio
B) Square root of two
C) Pi
D) Euler's number
- 15. Which sequence obeys the same recurrence relation as Fibonacci numbers and forms a complementary pair of Lucas sequences?
A) Bernoulli numbers
B) Prime numbers
C) Catalan numbers
D) Lucas numbers
- 16. What is the recurrence relation for Fn when n > 1?
A) Fn = Fn-1 + Fn-2
B) Fn = 2Fn-1
C) Fn = Fn-1 * Fn-2
D) Fn = Fn-1 - Fn-2
- 17. What is the relation between F-n and Fn for any integer n?
A) F-n = (-1)n+1 Fn
B) F-n = -Fn
C) F-n = (-1)n Fn
D) F-n = Fn
- 18. In Sanskrit prosody, how many units of duration does a long (L) syllable have?
A) 3 units
B) 4 units
C) 1 unit
D) 2 units
- 19. In Sanskrit prosody, how many units of duration does a short (S) syllable have?
A) 1 unit
B) 3 units
C) 2 units
D) 0.5 units
- 20. According to the Sanskrit poetic tradition, what is the number of patterns of duration m units?
A) Fm-1
B) Fm
C) 2Fm
D) Fm+1
- 21. Who authored the Natya Shastra, expressing knowledge of the sequence?
A) Hemachandra
B) Gopala
C) Bharata Muni
D) Virahanka
- 22. Whose work on the sequence is known only through a quotation by Gopala around 1135?
A) Fibonacci
B) Virahanka
C) Hemachandra
D) Pingala
- 23. Who wrote that 'the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta'?
A) Pingala
B) Hemachandra
C) Gopala
D) Virahanka
- 24. What problem did Fibonacci use to illustrate the sequence in Liber Abaci?
A) Growth of rabbit populations
B) Division of inheritance
C) Trading exchange rates
D) Speed of falling objects
- 25. In Fibonacci's rabbit problem, at what age do breeding pair of rabbits mate?
A) One month
B) Birth
C) Two months
D) Three months
- 26. In Fibonacci's rabbit problem, when do breeding pairs always produce another pair of rabbits?
A) Every week
B) At the end of their second month
C) At the end of their first month
D) After one year
- 27. Which 19th-century number theorist first used the name 'Fibonacci sequence'?
A) Abraham de Moivre
B) Édouard Lucas
C) Jacques Philippe Marie Binet
D) Daniel Bernoulli
- 28. What is the closed-form expression for Fibonacci numbers commonly known as?
A) Lucas's formula
B) Binet's formula
C) Fibonacci's identity
D) Kepler's formula
- 29. Who besides Jacques Philippe Marie Binet already knew the closed-form expression for Fibonacci numbers?
A) Johannes Kepler and Édouard Lucas
B) Pingala and Virahanka
C) Hemachandra and Bharata Muni
D) Abraham de Moivre and Daniel Bernoulli
- 30. What is the numerical value of phi (φ) to several decimal places?
A) 3.14159...
B) 1.61803...
C) 2.71828...
D) 0.61803...
- 31. What is the numerical value of psi (ψ), the conjugate of phi?
A) 0.61803...
B) 1.61803...
C) -0.61803...
D) -1.61803...
- 32. What quadratic equation do phi and psi satisfy?
A) x2 + x - 1 = 0
B) x2 - x + 1 = 0
C) x2 + x + 1 = 0
D) x2 - x - 1 = 0
- 33. Which initial values of Un generate the sequence of Lucas numbers using the general solution formula with a = 1?
A) U0 = 1 and U1 = 1
B) U0 = 0 and U1 = 1
C) U0 = 3 and U1 = 2
D) U0 = 2 and U1 = 1
- 34. How can the n-th Fibonacci number Fn be found using rounding for n >= 0?
A) Fn = floor(φn / sqrt(5))
B) Fn = floor(φn / sqrt(5) + 0.5)
C) Fn = ceil(φn / sqrt(5))
D) Fn = round(φn / 2)
- 35. For what values of n is the rounding error in Binet's formula less than 0.1?
A) n >= 4
B) n >= 12
C) n >= 1
D) n >= 8
- 36. For what values of n is the rounding error in Binet's formula less than 0.01?
A) n >= 4
B) n >= 8
C) n >= 1
D) n >= 16
- 37. Approximately what is the number of digits in Fn asymptotic to?
A) n * sqrt(5)
B) n / 2
C) n * log10(2)
D) n * log10(φ) ≈ 0.2090 n
- 38. For every integer d > 1, how many Fibonacci numbers have d decimal digits?
A) 4 or 5
B) Exactly 5
C) 8 or 9
D) 1 or 2
- 39. Who observed that the ratio of consecutive Fibonacci numbers converges to the golden ratio?
A) Pingala
B) Johannes Kepler
C) Leonardo of Pisa
D) Jacques Philippe Marie Binet
- 40. What is the limit of Fn+m / Fn as n approaches infinity?
A) φm
B) φn
C) 1
D) m * φ
- 41. What identity does the golden ratio satisfy regarding its square?
A) φ2 = φ - 1
B) φ2 = 2φ
C) φ2 = φ + 1
D) φ2 = 1 - φ
- 42. What higher power decomposition formula yields Fibonacci numbers as linear coefficients?
A) φn = φ2 + Fn
B) φn = Fn φ + Fn-1
C) φn = Fn-1 φ + Fn
D) φn = Fn2 φ
- 43. A positive integer x is a Fibonacci number if and only if at least one of which two expressions is a perfect square?
A) 5x2 + 4 or 5x2 - 4
B) x2 + 5 or x2 - 5
C) 5x2 + 1 or 5x2 - 1
D) x2 + 4 or x2 - 4
- 44. What type of matrix is the 2x2 matrix A with a determinant of -1?
A) Zero matrix
B) Identity matrix
C) Unimodular matrix
D) Symmetric matrix
- 45. What equation represents Cassini's identity?
A) 1 = Fn+1 + Fn-1
B) (-1)n = Fn+1 Fn-1 - Fn2
C) Fn2 = Fn+12 + Fn-12
D) (-1)n = Fn2 - Fn+1 Fn-1
- 46. What is the value of F4?
A) 3
B) 1
C) 2
D) 5
- 47. What is the sum of the first Fibonacci numbers with odd index up to F2n-1?
A) F2n
B) F2n+1 - 1
C) F2n - 1
D) F2n+1
- 48. Which method is commonly used to easily prove Fibonacci identities according to the text?
A) Matrix inversion
B) Mathematical induction
C) Geometric proofs
D) Calculus integration
- 49. What is the name of the identity given by F_n2 - F_{n+1}F_{n-1} = (-1)^{n-1}?
A) Cassini's identity
B) Catalan's identity
C) d'Ocagne's identity
D) Lucas's identity
- 50. Which mathematician's name is associated with the identity F_mF_{n+1} - F_{m+1}F_n = (-1)nF_{m-n}?
A) Cassini
B) Fibonacci
C) d'Ocagne
D) Catalan
- 51. What symbol represents the golden ratio in the text?
A) \pi
B) \lambda
C) \varphi
D) \psi
- 52. What is the conjugate of the golden ratio denoted as in the text?
A) \gamma
B) \varphi
C) \psi
D) \omega
- 53. What method can be used experimentally to find doubling and tripling identities like F_{3n}?
A) Polynomial division
B) Mathematical induction
C) Lattice reduction
D) Integration by parts
- 54. What type of generating function of the Fibonacci sequence is derived from the recurrence relation in the text?
A) Logarithmic
B) Ordinary
C) Exponential
D) Polynomial
- 55. The recurrence relation yields what kind of differential equation?
A) Homogeneous linear differential equation
B) Partial differential equation
C) Non-linear differential equation
D) Non-homogeneous linear differential equation
- 56. What symbol represents the conjugate of the golden ratio in the text?
A) e
B) ψ
C) r
D) φ
- 57. What type of function is the exponential generating function of the Fibonacci numbers?
A) Entire function
B) Discontinuous function
C) Rational function
D) Trigonometric function
- 58. What letter represents the golden ratio in the text?
A) e
B) φ
C) ψ
D) r
- 59. In the exponential generating function derivation, F(0) is equal to which initial value?
A) F'_''(0)
B) F0
C) F2
D) F1
- 60. What mathematical functions are infinite sums over reciprocal Fibonacci numbers sometimes evaluated in terms of?
A) Trigonometric functions
B) Bessel functions
C) Theta functions
D) Gamma functions
- 61. Who proved that the reciprocal Fibonacci constant is irrational?
A) Richard André-Jeannin
B) Leonardo Fibonacci
C) Carl Friedrich Gauss
D) Leonhard Euler
- 62. What series gives the identity involving sums over powers of two indices of Fibonacci numbers?
A) Fourier series
B) Taylor series
C) Millin's series
D) Lambert series
- 63. What function is used in the expression for the sum of all even-indexed reciprocal Fibonacci numbers?
A) Riemann zeta function
B) Lambert series
C) Prime counting function
D) Fibonacci polynomial
- 64. Which number in the Fibonacci sequence is even according to the text?
A) Every fifth number
B) Every third number
C) Every fourth number
D) Every second number
- 65. Generally, every k-th number of the Fibonacci sequence is a multiple of which term?
A) F1
B) F3
C) Fk
D) F0
- 66. What type of sequence is the Fibonacci sequence considered to be based on its divisibility properties?
A) A divisibility sequence
B) A prime sequence
C) An arithmetic sequence
D) A geometric sequence
- 67. What function is used in the stronger divisibility property of the Fibonacci sequence?
A) Greatest common divisor
B) Least common multiple
C) Modulo operator
D) Legendre symbol
- 68. Any three consecutive Fibonacci numbers have what property regarding each other?
A) They are all multiples of 5
B) They share a common divisor greater than 1
C) They are pairwise coprime
D) They are all even
- 69. If a prime p is congruent to 1 or 4 modulo 5, what does p divide?
A) F3
B) Fp
C) Fp-1
D) Fp+1
- 70. If the prime p equals 5, which Fibonacci number does it divide?
A) Fp-1
B) Fp
C) Fp+1
D) F1
- 71. What mathematical symbol is used to combine the prime divisibility cases into a single non-piecewise formula?
A) Product symbol
B) Integral symbol
C) Legendre symbol
D) Summation symbol
- 72. What is the fourth Fibonacci prime listed in the text?
A) 233
B) 89
C) 5
D) 13
- 73. What is the only nontrivial square Fibonacci number?
A) 8
B) 55
C) 1
D) 144
- 74. Who proved in 2001 that there is only a finite number of perfect power Fibonacci numbers?
A) Luo Ming
B) Y. Bugeaud
C) Attila Pethő
D) Vern Hoggatt
- 75. Who conjectured that 1, 3, 21, and 55 are the only triangular Fibonacci numbers?
A) Y. Bugeaud
B) Vern Hoggatt
C) Luo Ming
D) Attila Pethő
- 76. Who proved the conjecture regarding triangular Fibonacci numbers?
A) Luo Ming
B) Attila Pethő
C) M. Mignotte
D) Vern Hoggatt
- 77. Can any Fibonacci number be a perfect number?
A) No
B) Yes, the number 144
C) Yes, the number 28
D) Yes, the number 6
- 78. What method is used to calculate the matrix power Am?
A) Prime factorization
B) Matrix addition
C) Modular exponentiation
D) Gaussian elimination
- 79. Which theorem states that almost every Fibonacci number has a prime factor not found in any smaller Fibonacci number?
A) Wilson's theorem
B) Fermat's theorem
C) Euler's theorem
D) Carmichael's theorem
- 80. What is the value of F8?
A) 21
B) 13
C) 34
D) 55
- 81. What is the value of F10?
A) 34
B) 55
C) 21
D) 89
- 82. What is the value of F11?
A) 34
B) 144
C) 89
D) 55
- 83. Primes p for which F_{p - (p/5)} is congruent to 0 modulo p2 are called what?
A) Wall–Sun–Sun primes
B) Sophie Germain primes
C) Mersenne primes
D) Fermat primes
- 84. Up to which index i are all known factors of Fibonacci numbers F(i) collected in repositories?
A) 1000
B) 100000
C) 50000
D) 10000
- 85. What is the value of F13?
A) 89
B) 144
C) 233
D) 377
- 86. What is the value of F14?
A) 233
B) 377
C) 610
D) 144
- 87. What is the value of F15?
A) 377
B) 233
C) 987
D) 610
- 88. What are the lengths of the periods formed when Fibonacci members are taken modulo n called?
A) Lucas periods
B) Pisano periods
C) Fibonacci periods
D) Modular periods
- 89. What are the negafibonacci numbers produced by?
A) Generalizing the index to negative integers
B) Adding 3 numbers together
C) Starting with other integers
D) Generalizing the index to real numbers
- 90. What type of sequence uses the Fibonacci recursion with other starting points to generate sequences where all numbers are composite?
A) Primefree sequences
B) Pell numbers
C) Padovan sequence
D) Lucas numbers
- 91. What are the defining recurrence numbers for Pell numbers?
A) Pn = 2Pn-1 + Pn-2
B) Pn = Pn-1 + Pn-2
C) Pn = Pn-2 + Pn-3
D) Pn = 3Pn-1 + Pn-2
- 92. The number of ways to climb a staircase of 5 steps taking one or two steps at a time is equal to which Fibonacci number?
A) F6
B) F5
C) F4
D) F7
- 93. What is the number of binary strings of length n without an even number of consecutive 0s or 1s?
A) Fn+2
B) Fn+1
C) Fn
D) 2Fn
- 94. Who was able to show that Fibonacci numbers can be defined by a Diophantine equation, solving Hilbert's tenth problem?
A) Zeckendorf
B) Fibonacci
C) Binet
D) Yuri Matiyasevich
- 95. What is a sum of Fibonacci numbers satisfying Zeckendorf's theorem called?
A) A Pascal representation
B) A Zeckendorf representation
C) A modular representation
D) A Fibbinary representation
- 96. Starting with 5, every second Fibonacci number is the length of what part of a right triangle with integer sides?
A) The base
B) The hypotenuse
C) The shortest side
D) The perimeter
- 97. In which lemma do Fibonacci numbers appear, used to prove connections between the circle packing theorem and conformal maps?
A) The Zeckendorf lemma
B) The cycle lemma
C) The modular lemma
D) The ring lemma
- 98. What algorithmic runtime analysis uses consecutive Fibonacci numbers as the worst case input?
A) Dijkstra's algorithm
B) Merge sort algorithm
C) Quick sort algorithm
D) Euclid's algorithm
- 99. A Fibonacci tree is considered which type of specific tree with the fewest nodes for a given height?
A) AVL tree
B) Spanning tree
C) Decision tree
D) Syntax tree
- 100. Which audio file format uses the Fibonacci number series for optional lossy compression?
A) MP3
B) FLAC
C) IFF 8SVX
D) WAV
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