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l-1j-cho
- 104
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Does anyone have a DIRECT proof of the relationship between Pascal Triangle and Fibonacci Sequence? I mean not like induction or other method of proof but a direct method. I try to google it but couldn't find one
http://en.wikipedia.org/wiki/Pascal_triangle#Overall_patterns_and_properties"One property of the triangle is revealed if the rows are left-justified. In the triangle below, the diagonal coloured bands sum to successive Fibonacci numbers.
The Pascal Triangle is a famous mathematical pattern named after the French mathematician Blaise Pascal. It is an infinite triangular array of numbers where the first and last numbers in each row are always 1, and each subsequent number is the sum of the two numbers directly above it.
The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The sequence is named after the Italian mathematician Leonardo Fibonacci, who introduced it to the Western world in his book Liber Abaci.
The relationship between the Pascal Triangle and Fibonacci Sequence is that the sum of the numbers in each diagonal of the Pascal Triangle is equal to the corresponding number in the Fibonacci Sequence. Additionally, the ratio between adjacent numbers in the Fibonacci Sequence approaches the golden ratio, which is also found in the Pascal Triangle.
The Pascal Triangle is commonly used in probability and combinatorics to calculate the number of combinations or outcomes in a given scenario. The numbers in the Pascal Triangle represent the possible outcomes of a binomial experiment, where each row corresponds to a different number of trials and each number in the row represents the number of successful outcomes.
The Pascal Triangle and Fibonacci Sequence have various real-life applications, such as in finance, architecture, and computer science. In finance, these patterns can be used to predict stock market trends and interest rates. In architecture, they can be used to create aesthetically pleasing designs. In computer science, they are used in data compression and encryption algorithms.