#### Presentation Title

The Fibonacci Sequence and its Linear Recurrence Properties

November 2016

November 2016

HUB 302-#140

Poster

#### Abstract

The Fibonacci numbers have been observed in nature, like in the growth patterns of plants, and have been studied in mathematics since their appearance in the Liber Abaci in the 13th century. The Fibonacci numbers are a particular example of a mathematical object called a linear recurrence sequence, where the next term in the sequence is created by adding the two previous terms together. The sequence, following with the pattern 1, 1, 2, 3, 5, 8, …, is defined by Fn, starting with F1= 1, F2= 1, and for all n ≥ 3, Fn = 1Fn-1 + 1Fn-2 . We chose a formula often used with the Fibonacci numbers, and applied it to a general form of the Fibonacci sequence, En . The Fibonacci sequence was changed by substituting constants a and b to form a new recurrence relation, defined En= aEn-1+bEn-2 , and we investigated the patterns found in different sequences, with various values of a and b. We formed new sequence, Bn, with a=1 and b=2, and the sequence beginning with B0 = 0 and B1 = 1. This became the formula Bn= 1Bn-1 + 2Bn-2 for all n ≥ 2. When the pattern was subjected to the formula (Bn)2 - (Bn-1)(Bn+1), a consecutive series of powers of (-2) emerged, fitting the pattern (Bn)2 - (Bn-1)(Bn+1) = (-2)n-1. Similar patterns were also found in other sequences, leading us to hypothesize that the general formula would fit the pattern (En)2 - (En-1)(En+1) = (-b)n-1 . This meant that the changes in the pattern were caused by changing the constant b in the linear recurrence sequence formula. We were able to then prove this hypothesis by the method of mathematical induction.

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The Fibonacci Sequence and its Linear Recurrence Properties

HUB 302-#140

The Fibonacci numbers have been observed in nature, like in the growth patterns of plants, and have been studied in mathematics since their appearance in the Liber Abaci in the 13th century. The Fibonacci numbers are a particular example of a mathematical object called a linear recurrence sequence, where the next term in the sequence is created by adding the two previous terms together. The sequence, following with the pattern 1, 1, 2, 3, 5, 8, …, is defined by Fn, starting with F1= 1, F2= 1, and for all n ≥ 3, Fn = 1Fn-1 + 1Fn-2 . We chose a formula often used with the Fibonacci numbers, and applied it to a general form of the Fibonacci sequence, En . The Fibonacci sequence was changed by substituting constants a and b to form a new recurrence relation, defined En= aEn-1+bEn-2 , and we investigated the patterns found in different sequences, with various values of a and b. We formed new sequence, Bn, with a=1 and b=2, and the sequence beginning with B0 = 0 and B1 = 1. This became the formula Bn= 1Bn-1 + 2Bn-2 for all n ≥ 2. When the pattern was subjected to the formula (Bn)2 - (Bn-1)(Bn+1), a consecutive series of powers of (-2) emerged, fitting the pattern (Bn)2 - (Bn-1)(Bn+1) = (-2)n-1. Similar patterns were also found in other sequences, leading us to hypothesize that the general formula would fit the pattern (En)2 - (En-1)(En+1) = (-b)n-1 . This meant that the changes in the pattern were caused by changing the constant b in the linear recurrence sequence formula. We were able to then prove this hypothesis by the method of mathematical induction.