LARA.

ALLAN ANDREI E
MATHMOW
RECTO

1. B
2. C
3. B
4. B
5. B
6. B
7. C
8. B
9. D
10. B
11. B
12. B
13. D
14. B
15. B
16. C
17. B
18. B
19. C
20. B
21. B
22. D
23. D
24. C
25. B
26. B
27. A
28. A
29. A
30. B
31. A
32. B
33. C
34. B
35. B
36. A
37. B
38. B
39. D
40. A
 41. B
42. B
43. C
44. C
45. B
46. B
47. A
48. A
49. B
50. B

II

1. The Fibonacci sequence, named after Leonardo of Pisa, also known as Fibonacci, is a
series of numbers where each number is the sum of the two preceding ones. This
sequence was first introduced in Fibonacci's book "Liber Abaci" in 1202.Fibonacci
discovered the sequence while trying to solve a problem involving the growth of a
population of rabbits. The problem described a scenario where a pair of rabbits produces
another pair every month, assuming that each pair takes a month to mature and start
reproducing. The sequence emerged as Fibonacci counted the number of pairs of
rabbits over several months, leading to the famous sequence: 0, 1, 1, 2, 3, 5, 8, 13, and
so on.In modern times, the Fibonacci sequence has found numerous applications across
various fields. In computer science, the sequence is used in algorithms, such as the
Fibonacci search technique and dynamic programming. In nature, the Fibonacci
sequence appears in the arrangement of leaves, petals, and branches in plants, as well
as in the spiral patterns of shells and hurricanes.Moreover, artists and architects have
incorporated the Fibonacci sequence into their work to achieve aesthetically pleasing
proportions and compositions. In finance, the sequence is utilized in technical analysis to
predict market trends and support trading decisions. Overall, the Fibonacci sequence's
mathematical properties and aesthetic appeal continue to inspire innovation and
creativity in diverse disciplines.

2. The Fibonacci sequence exhibits fascinating mathematical properties that stem from its
recursive nature, where each term is derived from the sum of the two preceding terms.
Mathematically, the sequence is defined as follows:F(n) = F(n-1) + F(n-2) for n > 1,
with the base cases F(0) = 0 and F(1) = 1.This recursive formula generates the sequence: 0, 1,
1, 2, 3, 5, 8, 13, and so on. Each term is the sum of the two preceding terms, showcasing the
sequence's self-replicating pattern.The Fibonacci sequence is closely linked to the concept of
the golden ratio, denoted by the Greek letter phi (φ). The golden ratio is approximately 1.618
and is derived from the ratio of consecutive Fibonacci numbers as the sequence progresses. As
the terms in the Fibonacci sequence increase, the ratio between consecutive terms approaches
the golden ratio.In nature, the Fibonacci sequence and the golden ratio manifest in various
ways. For instance, the spiral patterns of sunflowers, pinecones, and the arrangement of seeds
in a sunflower follow Fibonacci numbers and the golden ratio. In human-made structures,
architects and designers use the golden ratio to create visually appealing proportions in
buildings, artwork, and even musical compositions.The mathematical beauty of the Fibonacci
sequence and its connection to the golden ratio highlight the harmony and symmetry found in
nature and human creativity, showcasing the universality of these mathematical principles
across different domains.

3. The Fibonacci sequence's presence in nature is a testament to its mathematical

elegance and prevalence in biological settings. One of the most striking examples of the
Fibonacci sequence in nature is the arrangement of leaves on a stem. Many plants
exhibit phyllotaxis, a pattern where leaves are positioned in a spiral formation that
follows Fibonacci numbers. This arrangement allows each leaf to receive optimal
sunlight and nutrients without shading others, promoting efficient photosynthesis.Another
notable example is the pattern of seeds in a sunflower head. The seeds are arranged in
spirals that follow Fibonacci numbers, with the number of spirals typically being
consecutive Fibonacci numbers. This arrangement maximizes seed packing efficiency
and ensures each seed is evenly spaced for optimal growth and distribution.In the case
of mollusks, such as the nautilus and various snail species, the Fibonacci sequence is
reflected in the spiral shells. The growth of the shell follows a logarithmic spiral that
approximates the Fibonacci spiral, providing structural strength and efficient use of
space as the organism grows.These Fibonacci patterns in nature are believed to be
evolutionarily advantageous for several reasons. The optimal spacing and arrangement
of leaves, seeds, and shells allow organisms to maximize exposure to resources like
sunlight, water, and nutrients. Additionally, these patterns may aid in seed dispersal,
pollination, and protection from predators by enhancing the organism's survival and
reproductive success.Overall, the presence of the Fibonacci sequence in nature
underscores the efficiency and effectiveness of mathematical principles in shaping
biological structures and behaviors. The patterns observed in plants and animals
highlight the intricate relationship between mathematics and evolution, showcasing the
beauty and functionality of natural systems.

4. The Fibonacci sequence and the golden ratio have long been utilized in art and architecture
to create aesthetically pleasing compositions and structures that resonate with viewers. These
mathematical principles have inspired artists and architects to achieve harmony, balance, and
beauty in their creations.
In art, renowned artists like Leonardo da Vinci and Salvador Dali incorporated the golden ratio
and Fibonacci sequence in their works. Da Vinci's "Vitruvian Man" is a classic example of the
golden ratio's influence, as the proportions of the human figure align with this mathematical
ratio, symbolizing balance and perfection. Dali's paintings, such as "The Sacrament of the Last
Supper," feature spirals and proportions derived from the Fibonacci sequence, adding a sense
of order and elegance to his surrealistic style. In architecture, the Parthenon in Greece is a
notable example of the golden ratio's application. The temple's dimensions and proportions
adhere to the golden ratio, creating a sense of visual harmony and balance that enhances its
grandeur. Similarly, the Notre-Dame Cathedral in Paris and the Taj Mahal in India feature
architectural elements based on Fibonacci numbers, such as spiral staircases and domes,
adding a sense of rhythm and symmetry to the structures. The use of the Fibonacci sequence
and the golden ratio in art and architecture not only enhances the aesthetic appeal but also
provides structural benefits. These mathematical principles help create visually pleasing
proportions, symmetry, and balance that evoke a sense of order and beauty. Additionally,
incorporating these patterns can improve structural stability, acoustics, and lighting in buildings,
enhancing the overall experience for occupants and visitors. Overall, the integration of the
Fibonacci sequence and the golden ratio in art and architecture showcases the timeless allure
and universality of these mathematical principles, demonstrating their ability to transcend
disciplines and inspire creativity and innovation.

5. The Fibonacci sequence has made a significant impact on modern science and technology,
particularly in the fields of algorithms, data structures, and computer graphics. Its recursive
nature and mathematical properties have inspired innovative applications that enhance
efficiency and performance in various technological domains.

In algorithms, the Fibonacci sequence is utilized in dynamic programming and optimization

problems. The Fibonacci search technique, for example, leverages the sequence's self-
replicating pattern to efficiently search for a target value in a sorted array. By utilizing the
Fibonacci numbers as indices for comparison, this algorithm reduces the number of
comparisons required, leading to faster search times.

In data structures, the Fibonacci heap is a specialized data structure used in graph algorithms
and optimization problems. Unlike traditional heaps, the Fibonacci heap maintains a set of trees
with a specific structure that allows for efficient operations like insertions, deletions, and
decrease key operations. This data structure's amortized time complexity is superior to other
heap structures, making it ideal for applications requiring frequent updates and operations.

In computer graphics, the Fibonacci sequence is applied in generating visually appealing

patterns and textures. Artists and designers use Fibonacci spirals and golden rectangles to
create harmonious compositions and layouts in digital art and graphic design. These
mathematical principles help achieve balanced and aesthetically pleasing visuals that captivate
viewers and enhance user experience.

Specific examples of how the Fibonacci sequence is utilized in contemporary technological

advancements include image compression algorithms, where Fibonacci coding techniques are
employed to reduce file sizes while maintaining image quality. Additionally, in scientific
research, the Fibonacci sequence is used in modeling natural phenomena, such as population
dynamics, gene sequences, and fractal patterns, to analyze complex systems and predict
outcomes.

Overall, the Fibonacci sequence's influence on modern science and technology underscores its
versatility and applicability across diverse domains, driving innovation and efficiency in algorithm
design, data structures, computer graphics, and scientific research.