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Beginning Activity Beginning Activity 2: The Fibonacci Numbers
A4US

The Fibonacci numbers are a sequence of natural numbers \(f_1 ,f_2 ,f_3 , \ldots ,f_n , \ldots\) defined recursively as follows:

  • \(f_1 = 1\) and \(f_2 = 1\text{,}\) and

  • For each natural number \(n\text{,}\) \(f_{n + 2} = f_{n + 1} + f_n\text{.}\)

In words, the recursion formula states that for any natural number \(n\) with \(n \geq 3\text{,}\) the \(n^{th}\) Fibonacci number is the sum of the two previous Fibonacci numbers. So we see that

\begin{align*} f_3 \amp = f_2 + f_1 = 1 + 1 = 2,\\ f_4 \amp = f_3 + f_2 = 2 + 1 = 3, \text{ and }\\ f_5 \amp = f_4 + f_3 = 3 + 2 = 5\text{.} \end{align*}

1.

Calculate \(f_6\) through \(f_{20}\text{.}\)

2.

Which of the Fibonacci numbers \(f_1\) through \(f_{20}\) are even? Which are multiples of 3?

3.

For \(n = 2\text{,}\) \(n = 3\text{,}\) \(n = 4\text{,}\) and \(n = 5\text{,}\) how is the sum of the first \((n - 1)\) Fibonacci numbers related to the \((n + 1)^{st}\) Fibonacci number?

4.

Record any other observations about the values of the Fibonacci numbers or any patterns that you observe in the sequence of Fibonacci numbers. If necessary, compute more Fibonacci numbers.