Garden 3:
Fibonacci Sequence

• Recursion Formula

This is the most exciting part of the journey! We'll be exploring an amazing number pattern called the Fibonacci Sequence. Before we begin, though, let's take a look at what we call a recursion formula.

"A recursion formula allows the term of a sequence to be determined from the previous."

t_n = t_n-1+3, t₁=2

A recursion formula states how "t_n" is related to the previous term "t_n-1". It then tells you what the first term of the sequence is to give you the starting point.

You are then able to list the term of the sequence using the formula. Remember, that the first term is already given:

t₁=2

The term before t₂ is t₁, which, according to our recursion formula, is 2. Now it's all simple substitution!

t₂=t₁+3
=2+3
=5

• Fibonacci Sequence

"The Fibonacci Sequence, developed by Italian mathematician Fibonacci, can be described by the following recursion formula:"

t₁=t₂=1
t_n=t_n-1+t_n-2

Using the recursion formula, the Fibonacci Sequence looks like this:

t₁=1
t₂=1
t₃=t₂+t₁
  =1+1
  =2
t₄=1+2
  =3
t₅=2+3
  =5
t₆=3+5
  =8
t₇=5+8
  =13
t₈=8+13
  =21

What have you noticed? In short, the Fibonacci Sequence begins with 1 and consists of the sum of the 2 previous terms.

{1,1,2,3,5,8,13,21,34,55,89,144,233,...}

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• Garden Highlight

There are many cool features to the Fibonacci Sequence. Check this out:

1. Let's add the first 5 terms together and add an extra 1

1+1+2+3+5+1=13

...We get the 7^th term of the sequence!

Let's try adding the first 6 and add 1

1+1+2+3+5+8+1=21

...It's the 8^th term of the sequence!!
The pattern will always work as long as we start at the beginning--"1". In other words, we can add the first 13 terms and 1 and we'll get the 15^th term, the first 77 and 1 and get the 79^th and so on.

However, we can also skip some numbers, except that now we'll be getting the term right after one of which we stop--DO NOT forget the extra 1 though:

1+3+8+21+1=34

And so on.

2. ...There's still more! Check out the following:

3² = 9
5² = 25
9 + 25 = 34

...The sum of square of a term and the square of the term after it gives us another Fib term!

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Fibonacii Sequence