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RJ Emery
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While a Fibonacci sequence is the sum of the previous two terms, what of sums of the preceeding n terms, and have such sequences (n > 2) been found to occur in the natural world?
Thank you for the references.ktoz said:Lots of info online. Fibonacci sequence http://en.wikipedia.org/wiki/Fibonacci_number"
A Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers. It starts with 0 and 1, and continues by adding the two previous numbers to get the next number in the sequence. For example, the first few numbers in a Fibonacci sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, etc.
The Fibonacci sequence is found in many natural phenomena, such as the branching of trees, the arrangement of leaves on a stem, and the spiral pattern of a nautilus shell. This is because the ratio between any two consecutive numbers in the sequence is approximately equal to the golden ratio, which is a mathematical relationship that appears frequently in nature.
The sum of all the numbers in a Fibonacci sequence up to a certain point is equal to the next number in the sequence minus 1. For example, the sum of the first 5 numbers in the sequence (0, 1, 1, 2, 3) is 7, which is equal to the next number (5) minus 1. This relationship continues as the sequence gets longer.
The relationship between the sums of preceding terms and Fibonacci sequences has practical applications in fields such as computer science, finance, and art. In computer science, the Fibonacci sequence is used for efficient searching and sorting algorithms. In finance, it can be used in predicting stock market trends. In art, the Fibonacci sequence is often used to create aesthetically pleasing compositions and designs.
Yes, there are some limitations and exceptions to this relationship. In some cases, the sum of the preceding terms may not equal the next number in the sequence minus 1. For example, if we start with 0 and 1, the next number in the sequence is 1, but the sum of the preceding terms (0 and 1) is also 1. Additionally, as the sequence gets longer, the relationship may become less accurate due to rounding errors and other factors.