VCE General Mathematics 1&2 - 2020 Edition
topic badge
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

What is Mathspace

About Mathspace