Fibonacci Sequence: Lesson 2

The Fibonacci Sequence: Review

In the previous lesson we generated a number sequence based on the following rules:

• It takes one month for two baby rabbits to grow to adulthood.
• It takes one month for a pair of adult rabbits to have a pair of baby rabbits.
• An adult pair has one pair of rabbits, one male, one female.

These rules are used to generate a number pattern. It doesn’t necessarily represent how actual rabbits reproduce.

This is what it looks like in May.

This is the number sequence generated.

Let’s use function notation to better understand this number sequence. Look at this table.

Starting at F(3), we know the following:

F(3) = F(2) + F(1)

You can verify this using the values in the table. Here are some other, similar equations. Verify each using the values in the table.

F(4) = F(3) + F(2)

F(5) = F(4) + F(3)

Do you see a pattern? What do you think the value of F(n) would be? The n represents the nth term in the Fibonacci sequence.

Following the pattern, we get this equation:

F(n) = F(n - 1) + F(n - 2)

This is known as a recursive formula. In a recursive formula, previous terms in the sequence are used to generate the next terms in the sequence. The Fibonacci Sequence is an example of a recursive formula.

Visualizing the Fibonacci Sequence

We can use a grid to visualize the Fibonacci Sequence. Starting with a blank grid, we’ll add square blocks that align with the numbers in the sequence. Take a look at this series of illustrations.

Study this visual pattern in more detail. In particular, make a note of the way the square blocks arrange themselves in a spiral around the original block.