NZ Level 7 (NZC) Level 2 (NCEA)
Fibonacci Sequence
Lesson

The Fibonacci sequence is named after the 13th Century mathematician Leonardo de Fibonacci of Pisa Italy.  It is a sequence that appears in nature - the branching of trees, the family tree of the honey-bee, the count of petals on flowers and many other instances.

Let's begin by looking at how to create a Fibonacci Spiral of our own.

Start with a blank piece of graph paper, and highlight a square with side length 1, somewhere towards the middle.

Next to that, highlight another square of side length 1.

Now, highlight a square of side length 2.  Use the two squares from before as one of the sides. (like this image).

Next, build a square of length 3 on the side of the squares we have already built, then a side of length 5 and another of length 8.  It should start to look like what I have here.

The final step is draw in a the spiral.  We construct the spiral but joining diagonally opposite corners in a swooping arc motion.  Start in the centre with the first square and then join corner to corner.  This may take a little bit of practice to get the arcs looking nice.

And so you have constructed a Fibonacci spiral.

The Fibonacci sequence is $1,1,2,3,5,8,13,21,34,55,89,...$1,1,2,3,5,8,13,21,34,55,89,... where, apart from the first two terms, each term is the sum of the previous two terms. It also possesses some unusual mathematical properties such as:

• The sum of the squares of any two consecutive terms is another Fibonacci term.
• The ratio of consecutive terms, namely $\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8}...$32,53,85,138... approaches the golden ratio where  $=$= $\frac{1+\sqrt{5}}{2}=1.618033...$1+52=1.618033...
• Apart from $F_1$F1$F_2$F2 and $F_4$F4 every prime term is sitting in a prime position. That is, if the $n$nth term  $F_n$Fn is prime then $n$n must be prime.

As an aside, the Fibonacci sequence is a member of a more general family of sequences which are all defined by how many terms are added to produce each new term. For example, the Tribonacci sequence begins $1,1,1,3,5,9,17,31,...$1,1,1,3,5,9,17,31,... where apart from the first three terms, each new term is formed as the sum of the previous three terms. The Tetranacci sequence begins $1,1,1,1,4,7,13,25,49,94,..$1,1,1,1,4,7,13,25,49,94,.. where the four previous terms are added.  It is interesting to note that the ratio of successive terms approaches a unique limiting value given by the real positive solution to a sequence of equations. For the Fibonacci sequence, we can show that the limiting value is phi given by the positive solution to the equation $x^2-x-1=0$x2x1=0. Similarly, the limiting value for the Tribonacci sequence is given by the positive solution to $x^3-x^2-x-1=0$x3x2x1=0. For the Tetranacci sequence the limiting value is given by the positive solution to $x^4-x^3-x^2-x-1=0$x4x3x2x1=0. The pattern continues indefinitely with all limiting values forming their own sequence of numbers that approach but never exceed $2$2

We can state the Fibonacci sequence using the recurrence relationship defined as:

$F_{n+2}=F_n+F_{n+1}$Fn+2=Fn+Fn+1 with the first two terms $F_1=1$F1=1,$F_2=1$F2=1

The rule for the $n$nth term is far more difficult to find, but in case you're curious, it is given by:

Worked Examples

QUESTION 1

Consider the following sequence.

$2,-1,1,0,1,1,\text{. . .}$2,1,1,0,1,1,. . .

1. Is the sequence a Fibonacci-type sequence (where each term is the sum of the two preceding terms)?

Yes

A

No

B

Yes

A

No

B
2. What are the next two terms of the sequence?

Write both terms on the same line, separated by a comma.

QUESTION 2

Use the fact that the Fibonacci sequence is defined by $t_n=t_{n-2}+t_{n-1}$tn=tn2+tn1, where $t_1=1$t1=1 and $t_2=1$t2=1, to generate terms $3$3 to $10$10.

Write all the values on the same line, separated by commas.

QUESTION 3

In the Fibonacci sequence, $t_{21}=10946$t21=10946, $t_{23}=28657$t23=28657 and $t_{24}=46368$t24=46368.

1. Find $t_{25}$t25.

2. Find $t_{19}+t_{20}$t19+t20.

3. Find $t_{22}$t22.

Outcomes

M7-3

Use arithmetic and geometric sequences and series

91258

Apply sequences and series in solving problems