Mathematics Sequences and Patterns

Study Notes

Arithmetic Sequence: Consists of terms with a constant difference; for example, a sequence like 5, 10, 15, 20 increases by 5.
Geometric Sequence: Each term is derived by multiplying the preceding term by a fixed number; an example is 2, 4, 8, 16, where each term is multiplied by 2.
Fibonacci Sequence: Each number is the sum of the two preceding numbers, starting typically with 1, 1, leading to 2, 3, 5, and beyond.
Harmonic Sequence: Composed of terms whose reciprocals form an arithmetic sequence.
Triangular Numbers: Formed by arranging dots to create an equilateral triangle; examples include 1, 3, 6, 10.

Golden Ratio: An irrational number approximately 1.618, prominent in nature, aesthetics, and architecture.
Symmetry: Indicates that an object remains unchanged under transformations such as reflection or rotation, highlighting balance in design.
Tessellation: Involves packing shapes tightly without gaps; a natural example is the honeycomb structure of beehives.
Pascal's Triangle: A triangular array showcasing binomial coefficients, useful in probability and algebraic expansions.

Fractal: A geometric shape exhibiting self-similarity, where smaller sections mirror the entire structure.
Self-Similarity: A defining characteristic of fractals; any portion resembles the overall shape.
Fractal Tree: Represents a natural fractal pattern, where smaller branches branch off from larger ones, demonstrating growth and pattern consistency.
Geodesic Sphere: A spherical structure composed of interconnected triangles, as observed in soap bubbles.

Exponential Growth: Describes a scenario where quantities grow rapidly at a constant rate relative to their current size; frequently modeled by geometric sequences.
Exponential Decay: Involves a rapid decrease of quantities at a constant rate, exemplified in phenomena like radioactive decay.

Isosceles Right Triangle: A specific triangle type with two equal-length sides and one right angle, foundational in trigonometric principles.
Euclidean Geometry: The study of flat space and shapes, forming the basis of classical geometry involving entities such as circles and squares.

Minimizing Surface Tension: A principle in nature observable in the formation of water droplets and cloud shapes, influencing how structures stabilize.
Logarithmic Spiral: A spiral pattern that enlarges as it circles, commonly found in natural forms like galaxies and seashells.
Pythagoras: An influential ancient Greek mathematician emphasizing the significance of numbers, laying groundwork for mathematical exploration and structural patterns.