![]() We now need only to show that the right hand side also counts these interlaced sequences. This means that terms of the sequence are not dependent on previous terms. Because of their lengths, given such a pair we can interlace their entries, forming an alternating sequence of digits and fence posts such as: $$1\, \square\, 0\, \square\, 1\, \blacksquare\, 1$$ We will call such sequences interlaced sequences. Binet’s Formula: The nth Fibonacci number is given by the following formula: fn (1 + 5 2)n (1 5 2)n 5 Binet’s formula is an example of an explicitly defined sequence. A Google search for a phrase like Fibonacci numbers in nature will produce a lot of hits. ![]() $$F_n=\frac\cdot F_n$, counts the number of ways of choosing a binary sequence of length $n-1$ and also a fence post coloring of length $n-2$. Precisely, + ' ' 5 + 5 ' + 5 The Fibonacci numbers can be pictured in a spiral of squares that fit neatly together: Fibonacci numbers have many interesting properties, and they frequently occur in patterns found in the natural world. One of the many, many things we investigated at the camp was the Fibonacci sequence, formed by starting with the two numbers $0$ and $1$ and then at each step, adding the previous two numbers to form the next: $$0,1,1,2,3,5,8,13,21,\ldots$$ If $F_n$ denotes the $(n+1)$st term of this sequence (where $F_0=0$ and $F_1=1$), then there is a remarkable formula for the $n$th term, known as Binet’s Formula: First, the terms are numbered from 0 onwards like this: So term number 6 is called 圆 (which equals 8). ![]() ![]() Some of the students mentioned that they felt even more inspired to study math further after our two-week program, but the inspiration went both ways – they inspired me with new ideas as well! The Rule The Fibonacci Sequence can be written as a 'Rule' (see Sequences and Series ). Consider the second-order recurrence ax n+2+bx n+1+cxn f. This year’s Prove it! Math Academy was a big success, and it was an enormous pleasure to teach the seventeen talented high school students that attended this year. The proofs are simple exercises, and it should be obvious how the theory extends to recurrences of other orders. ![]()
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