Binet's Formula For Research Paper

Example 1 Use Binet’s formula to determine the 10th, 25th, and 50th Fibonacci numbers.

Solution: Apply the formula with the aid of a scientific calculator and you will obtain the following:

F_10= 55, F_25= 75, 025, 〖 F〗_50= 1.258626902 × 〖10〗^10

The Fibonacci sequence is often evident in nature. The sunflower is an example. When we pay attention to the arrangement of seeds in its head, we can notice that they form spirals. In certain species, there are 21 spirals in the clockwise direction and 34 spirals in the counter clockwise direction.

In various species, you can count some with 34 and 55, or 55 and 89, or 89 and 144. This arrangement keeps the seeds uniformly packed regardless of how large the seed head is. The numbers 21,

When we take a closer look at a butterfly’s wings, a colorful mosaic is visible, as well as the waves of water and ripples by the sand dunes can be observable. The honeycomb and ice crystals give us an amazing evidence of symmetry. Furthermore, the living world is stuffed with streaked patterns of contrasting colors. These are only few of the many examples of the regularities in the world.

Patterns have underlying mathematical structures. Mathematics is indeed a search for regularities in the world and helps in the organization of patterns. Every living and non-living thing in the world may seem to follow a certain pattern on their own. As we look through the deeper sense in geometry and the mechanism of pattern formation, there are two classifications of patterns. These are self-organized patterns or inherent organizations and invoked organizations.

Self-Organized Patterns or Inherent

A little help from external connections is used to form a certain pattern.

The formation of a honeycomb is an invoked organization since the pattern is mainly caused by the bees. Another instance is the formation of a spider web when they produce a sticky orb, following a genetically determined recipe that creates radii and spirals of the web. In other words, invoked organized patterns needs something in order to function.

Mathematics provides a solution to pattern formation and organization. The geometry of pattern formations in nature can be linked mostly to mathematical numbers either directly or indirectly. However in some cases, while others may have been forced through, the high extent to which natural patterns follow a mathematical series and numbers is truly amazing. Furthermore, it is important for us to initially have an appreciable knowledge of a few mathematical series, ratios and