Notes on Sequences and Patterns

Patterns and Numbers in Nature

• Core mathematics topics: numeric patterns and geometric patterns.
• Patterns observed in art and nature illustrate how math appears in the real world.
• Ethnomathematics (see Page 9–10): the idea that artistry and abstraction in designs (e.g., textiles) reflect mathematical ingenuity and cultural heritage.

Fibonacci Sequence

• Definition and concept
  • A sequence where each term after the first two is the sum of the two preceding terms.
  • Recurrence relation: $F<em>n = F</em>{n-1} + F_{n-2} \, for \, n \,\ge 3$
• Common initial terms (one common convention): $F<em>0 = 0, \; F</em>1 = 1$ or $F<em>1 = 1, \; F</em>2 = 1$
• Example sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, …
• Significance: appears in biological settings, art, and nature; also linked to the golden ratio as the ratio of consecutive Fibonacci numbers approaches $\phi = \frac{1+\sqrt{5}}{2} \approx 1.618\dots$

The Golden Ratio

• Definition: $\phi = \frac{1+\sqrt{5}}{2} \approx 1.618\dots$
• Appearances in art and architecture:
  • ART: Da Vinci's Mona Lisa; Dali's Sacrament of the Last Supper.
  • ARCHITECTURE: Great Pyramid of Giza; Parthenon.

Link of Mathematics to Other Fields

• Ethnomathematics connects math to culture and traditional design (as above).
• Mathematics in chemistry and physics:
  • Graph theory to model molecular bonds.
  • Group theory to study crystal structures.
  • Linear algebra to characterize molecules.
• In biology and medicine:
  • Biostatistics, bioscience, medical research rely on math models.
  • Physiological genomics and mathematical models (e.g., PDE systems) for healing processes and tumor growth in fluid-like tissues.
• A fib (a Fibonacci poem):
  • A fib is a poem in which the number of syllables per line follows the Fibonacci sequence.
  • Invented by Greg Pincus; gained popularity and coverage (e.g., New York Times).
• Poetry excerpts (illustrative): short lines illustrating the integration of math-like structure into language (excerpted lines provided in the slides).

The Electromagnetic Spectrum, Light, and Waves

• Link to math and patterns in nature:
  • Relates to waves and frequencies; math helps describe wave behavior, interference, and propagation.
• Electromagnetic spectrum overview (visual references in slides, simplified):
  • Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.
  • Applications: AM/FM radio, TV, radar, WiFi, visible light, medical imaging, etc.
• Light waves and their sizes (visible spectrum reference):
  • Violet: ~$\lambda \approx 400\,\text{nm}$
  • Indigo: ~$\lambda \approx 425\,\text{nm}$
  • Blue: ~$\lambda \approx 470\,\text{nm}$
  • Green: ~$\lambda \approx 550\,\text{nm}$
  • Yellow: ~$\lambda \approx 580-600\,\text{nm}$
  • Orange: ~$\lambda \approx 630\,\text{nm}$
  • Red: ~$\lambda \approx 665\,\text{nm}$

Radiation Levels and Safety

• Non-ionizing vs Ionizing radiation:
  • Non-ionizing radiation sources (lower energy): power lines, mobile phones, microwaves, televisions, daylight, smart meters, WiFi, radios, baby monitors, remote controls, tanning beds, computers.
  • Ionizing radiation sources (higher energy): X-rays, gamma rays.
• Practical framing from slides: non-ionizing radiation spans everyday devices and comfort-level sources; ionizing radiation includes X-rays and gamma rays.

Mathematics in Chemistry and Physics (Fields of application)

• Graph theory, group theory, and linear algebra contribute to understanding molecular bonding, crystal structures, and molecular characterization.
• PDE-based models in physiology and medicine support healing processes and cancer tumor growth modeling in fluid-like tissues.

Inspiration from Nature and Art (Patterns and Growth)

• Patterns in nature encourage exploration of numerical sequences and recursive rules, linking to real-world patterns and aesthetic design.
• The study of sequences bridges observation (nature, art) with formal rules and proofs.

Notations and Definitions for Sequences

• Sequence: a list of numbers arranged in a specific order according to a rule. Example: $3, 6, 12, 24, \dots$
• Term: each number in a sequence. Example: in the sequence above, the terms are 3, 6, 12, 24, …
• Position (Index): the place of a term; counted as 1st, 2nd, 3rd, … term.
• Notation: $a<em>1$ denotes the first term, $a</em>2$ the second term, etc. In general, $a_n$ denotes the nth term.
• Rule (Pattern Rule): describes how terms are formed or how they change from one term to the next. Example: "Add 2" in the sequence $2, 4, 6, 8, \dots$
• Fibonacci Sequence (recalled): $F<em>0=0, F</em>1=1, \; F<em>n = F</em>{n-1} + F<em>{n-2}\; (n\ge2)$ or the variant with $F</em>1=1, F_2=1$

Classification of Sequences (by number of terms)

• Finite Sequence:
  • Has a specific number of terms; ends after a last term.
  • Example: $2, 4, 6, 8, 10$ (5 terms, ends at 10).
• Infinite Sequence:
  • Has no end; continues forever with the same rule.
  • Notation often uses ellipsis: $1, 2, 4, 8, 16, 32, \dots$

Common Examples of Sequences

• Arithmetic: constant difference between consecutive terms. Example: $2, 6, 10, 14, 18, \dots$ with common difference $d=4$.
• Geometric: constant ratio between consecutive terms. Example: $3, 6, 12, 24, 48, \dots$ with ratio $r=2$.
• Harmonic: reciprocals of an arithmetic sequence (classic harmonic sequences have the form $1/a<em>n$ where $a</em>n$ is arithmetic). (Note: slide lists a harmonic example as "4 ½ 10 13 6" which appears garbled; standard harmonic example structure is included here for context.)
• Fibonacci: recap above; appears again as a fundamental sequence with recursive rule.

Let’s Try: Practice problems and solutions

• Problem 1: 81, 27, 9, 3, ___
  • Answer: 1
  • Rule: divide the previous term by 3 to get the next term.
• Problem 2: (Example sequence not fully shown in slides; solution given on slides)
  • Answer: 10
  • Rule: each term is the sum of the two previous terms.
• Problem 3: 7, 14, 28, 56, 112
  • Answer: 112
  • Rule: multiply by 2.
• Problem 4: 10, 21, 31, ___
  • Answer: 52
  • Rule: next term is the sum of the two previous terms (i.e., 21+31=52).
• Problem 5: Sequence that matches "Squares of whole numbers" pattern
  • Sequence: 1, 4, 9, 16, 25, …
  • Next term: 25
  • Rule: Squares of whole numbers (i.e., $n^2\, (n=1,2,3,…)$).
• Problem 6: 3, 5, 9, 17, 33, ___
  • Answer: 65
  • Rule: each term is obtained by adding the next power of 2 to the previous term (i.e., add $2^{k}$ with k increasing: 2, 4, 8, 16, 32, …).