Emanfatimahh
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2 hours ago October 22, 2024
Roll number 24021 , 24032, 24054, 24003, 25052 has worked on collecting data . 24021 and 24032 has worked on never repeat itself part, roll number 24052 and 24003 has worked on uses and 24054 worked on examples and we have gathered data for your presentations , we will work on it and comile it according to our presentation.
Never Repeat Itself Slide 1: Why the Fibonacci Sequence Never Repeats Unique Formation of Terms • The sequence is generated by adding the two preceding terms, beginning with 0 and 1. • Each number is a unique result of the two previous values, so the exact repetition of values doesn’t occur after the initial terms. Why “1” Repeats • The number "1" repeats in the Fibonacci sequence due to the specific starting values: F(0)=0 and F(1)=1. • By the sequence rule, F(2)= F(1) + F(0) =1, causing "1" to appear again as the second term. • Beyond this, each term is a unique sum of its two predecessors, leading to rapid growth and divergence. This prevents any further repetitions, making "1" the only repeated value in the sequence. Exponential Growth and Divergence • Each new term grows rapidly and diverges from earlier ones, making cycles impossible. Infinite Expansion • The sequence is infinite, expanding endlessly without repeating values.
Slide 2: Cyclical Properties vs. Value Repetition Patterns in Properties, Not in Exact Values • Last Digits: Repeat every 60 terms. • Digital Roots: Cycle every 24 terms. • Golden Ratio: Ratios between terms converge on the Golden Ratio (~1.618), connecting math to nature and art. Mathematical Proof Against Repetition • Modular arithmetic shows exact value repetition would require a loop, which is impossible due to the sequence’s structure. Natural Significance • Fibonacci numbers are found in nature: the spirals in shells, flower patterns, and tree branches. • The sequence reflects mathematical harmony in natural forms, bridging theory and the real world.
USES Slide 1: Everyday Applications of the Fibonacci Sequence Natural Patterns • Appears in leaf arrangements, flower petals, and spiral shells, representing efficient growth in nature. Art and Aesthetics • Guides design, art, and photography, leveraging proportions near the Golden Ratio for visual appeal. Financial Analysis • Used in stock market analysis for Fibonacci retracement levels, helping predict support and resistance points. Architecture and Design • Inspires proportions in buildings and layouts, adding balance and harmony to structures.
Slide 2: Advanced Applications of the Fibonacci Sequence Computer Science • Optimizes algorithms in sorting, searching, and data structures, enhancing computational efficiency. Mathematics and Theory • Used to study numerical properties and patterns, contributing to number theory and mathematical proofs. Medical Imaging • Applied in MRI and spiral scanning for efficient imaging techniques. Music and Rhythm • Structures musical compositions, creating balance in rhythm and harmony.
EXAMPLES Slide 1: Everyday Examples of the Fibonacci Sequence Nature’s Patterns • Found in leaf arrangements, flower petals, and pine cones for efficient growth. Human Anatomy • Ratios in finger bones and facial features reflect Fibonacci proportions. Art and Photography • Used in composition to create visually pleasing scenes and balance. Stock Market • Fibonacci retracement levels help predict support and resistance points.
Slide 2: Specialized Examples of the Fibonacci Sequence Computer Science • Optimizes algorithms and data structures for efficiency. Architecture • Inspires proportions in iconic buildings for balance and beauty. Medical Imaging • Guides spiral scanning in MRI and CT for effective imaging. Music • Structures rhythms and scales, enhancing harmony and balance.