VCE General Mathematics 1&2 - 2020 Edition
6.07 The Fibonacci sequence
Lesson

## What is the Fibonacci sequence?

The Fibonacci sequence is named after the 13th Century mathematician Leonardo de Fibonacci of Pisa, Italy. It is an infinite 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.

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. Therefore, the sequence can be defined by the following second-order linear recurrence relation.

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

Remember this recurrence relation can also be defined by $F_n$Fn by finding the two previous terms $F_{n-1}$Fn1 and $F_{n-2}$Fn2:

$F_n=F_{n-1}+F_{n-2}$Fn=Fn1+Fn2 with the same first two terms $F_1=1,F_2=1$F1=1,F2=1

The explicit rule for the $n$nth term is far more difficult to find. It is given by:

The sequence itself 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.

### Fibonacci-type sequences

A Fibonacci-type sequence is a sequence similar to the Fibonacci sequence. Each term is still the sum of the previous two terms, except the first two numbers are not $1$1 and $1$1, but rather some other two numbers. $2,2,4,6,10,16,26,...$2,2,4,6,10,16,26,... is an example of a Fibonacci-type sequence. The recursive rule is still $F_{n+2}=F_{n+1}+F_n$Fn+2=Fn+1+Fn, but in this example $F_1=2$F1=2 and $F_2=2$F2=2 instead of $1$1 and $1$1 respectively.

#### Practice questions

##### Question 1

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 2

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 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.

### Graphing the Fibonacci sequence

Consider the Fibonacci sequence, modelled by the following recursive rule

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

Above is a graph of the Fibonacci sequence for the first eight terms of the sequence. It is clear from the graph that this sequence is a non-linear recurrence relation. Looking at the graph, it could be mistaken for a geometric sequence, since the behaviour of the graph appears to be exponentially increasing. However, there is no multiplication occurring in the recursive rule, only previous terms being added together.

### Outcomes

#### AoS3.18

Define and explain key concepts in generation of the Fibonacci and similar sequences using a recurrence relation, tabular and graphical display, and apply a range of related mathematical routines and procedures

#### AoS3.19

Define and explain key concepts in use of Fibonacci and similar sequences to model and analyse practical situations, and apply a range of related mathematical routines and procedures