![]() ![]() We can know thatthe ratio would be the same as that of the Fibonacci Sequence because the termsare generated by the same rule. Though we do not follow the same dividing process by f(n-1). We can express this sequence usingthe symbol of functions, too. The construction of members of a sequence takes the rule of that of Fibonaccisequence. Let's make a new sequence with 1 and 5 as the first and second number of thesequence. X: the ratio of each pair of adjacent terms Similarly, the ratio of every third termis x3, and it can be solved by 2x + 1. If we callthe ratio of each pair of adjacent terms as x, then the ratio of every secondterm is x2. The ratios shows the Fibonacci ratios are in geometric progression. In particular, we can draw a chart for the ratios of adjacent Fibonaccinumbers.Īs further exploration, we can draw a chart for all ratios which is shown inthe chart. The datawill also show the Fibonacci numbers, the ratio of each pair of adjacent terms,every second term, every third term and so on. Let's make a table of the Fibonacci numbers by using a spreadsheet. We can get a number 1.62 which is very familiar to us. To get the exact roots of the equation, we use the quadratic formula for x2 = x+ 1. The positive root of the equation exists between 1.5 and 2. The Algebra Xpresser is used forthe graphs. We can look at thegraph of two functions, y = x2 and y = x + 1. ![]() To get the approximate idea of two roots of the equation. If the concept of limit is introduced and n becomes large, then the equationcan be expressed as the following. We can express the Fibonacci Sequence using symbol of functions.Īs n increases, we get the Fibonacci numbers. It must be atime-consuming work for every student. Let's construct the numbers in Fibonacci Sequence by the rule. In this paper, I will treat many different aspects of the Fibonacci Sequence bygeneralizing the relationship of numbers, making a data with spreadsheet,drawing a chart for the data, extending the ratio of each pair of adjacentterms to the golden ratio, examining the ratio of every second term, everythird term, and so on, and understanding the algebraic characteristics of theresults by examining with Algebra Xpresser. They will say that eachterm is the sum of the two preceding terms. When the numbers are given, students will be able to generate the rule of thesequence through examining several adjacent numbers. As technology comes into aclassroom students also need to learn many different mathematical topicsdealing with different technologies. A sequence which is called " FibonacciSequence " is one of many famous sequences. Working with different types of number arrays can be one of interestingactivities in a mathematics classroom. The Spreadsheet in Mathematics Explorations ![]() L( n) = L( n − 1) + L( n − 2) for n ≥ 2, with L(0) = 2 and L(1) = 1.Ī natural number that is abundant but not semiperfect.Write-up 5 : The Spreadsheep in Mahtematics Explorations,Fibonacci Sequence Φ( n) is the number of positive integers not greater than n that are coprime with n. ![]() The nth term describes the length of the nth run This is a list of notable integer sequences and their OEIS links. ![]()
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