exploring patterns and Fibonacci Sequence

Fibonacci sequence

is a series of numbers in which each number is the sum of the two preceding numbers, starting with 1 and 1(1, 1, 2,3, 5, 8, 13, 21, 34, 55,...)

recursive rule

is a process of finding the next term using the preceding terms.

sequence

is a succession of numbers that follow a specific rule or pattern.

Leonardo of Pisa

known as Fibonacci (meaning "son of Bonacci"), lived in Italy from around 1170 to 1250. In the early 1200s, he introduced the Fibonacci sequence. Although it spread widely in Europe during his time, this number pattern was known in Indian mathematics as early as 200 BCE.

fibonacci Spirals

can be constructed by arranging a sequence of squares whose side lengths correspond to Fibonacci numbers. Alternatively, each square's area can equal the sum of the areas of the two preceding squares.

Fibonacci sequence in Trees

can be shown using a simple model of how a plant might grow new branches.

Fibonacci Sequence in Pascal’s triangle

Is useful for calculating binomial coefficients, expanding binomials, computing combinations, and understanding probability.

Golden Ratio

represented by the symbol $\phi$ (read as "phi"). The _____ is approximately 1.618. It can be observed by dividing each Fibonacci term by its immediate predecessor. As the terms increase, the ratio approaches $\phi$.

Binet’s Formula

used to find the any term in the Fibonacci sequence without needing to calculate all the preceding ones.