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