The Fibonacci numbers are a unique sequence of numbers where each new term is found by adding the two previous terms.

The Fibonacci Numbers are as follows:

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...$

## Fibonacci Sequences

A Fibonacci sequence refers to any sequence where each new term is found by adding the previous two terms, given any two starting values.

### Example 1

Which of the following are Fibonacci sequences?

a) $1, 4, 5, 9, 14, ...$

Calculate whether each new term is found by adding the previous two terms.

$t_3=t_1+t_2$
$t_3=1+4$
$t_3=5$

Check that \$t_3\$ is indeed equal to 5 by checking with the question.

This is correct.

$t_4=t_2+t_3$
$t_4=4+5$
$t_4=9$

This is correct.

$t_5=t_3+t_4$
$t_5=5+9$
$t_5=14$

This is correct.

This is a Fibonacci sequence as each new term is found by adding the two previous terms.

b) $3, 7, 10, 13, 23, ...$

Calculate whether each new term is found by adding the previous two terms.

$t_3=t_1+t_2$
$t_3=3+7$
$t_3=10$

Check that \$t_3\$ is indeed equal to 10 by checking with the question.

This is correct.

$t_4=t_2+t_3$
$t_4=7+10$
$t_4=17$

$t_4$ is supposed to be 13.

This is wrong.

This is not a Fibonacci sequence as each new term is not found by adding the previous term.

## Second Order Difference Equations for a Fibonacci Sequence

Second order difference equations for  Fibonacci sequences follows the following equation:

$f_{n+2}=f_n+f_{n+1}$   given $f_1$ and $f_2$

### Example 2

Find the first five terms of the following Fibonacci sequence given by the second order difference equation:

$f_{n+2}=f_n+f_{n+1}$   $f_1=2$ $f_2=1$

The question defines the first two terms so use these in the second order difference equation to calculate the remaining terms.

$f_1=2$ and $f_2=1$
$f_{n+2}=f_n+f_{n+1}$

$f_{3}=f_1+f_2$
$f_3=2+1$
$f_3=3$

$f_4=f_2+f_3$
$f_4=1+3$
$f_4=4$

$f_5=f_3+f_4$
$f_5=3+4$
$f_5=7$

The first five terms of the sequence are 2, 1, 3, 4 and 7.

### Example 3

Find the value of $t_2$ given $t_1=5, t_4=7$ and $t_5=13$.

We are given two consecutive values so all we need to do is to work backwards to find $t_3$.

$t_3=t_5-t_4$
$t_3=13-7$
$t_3=6$

Now we know $t_3$ we can work backwards to find $t_2$.

$t_2=t_4-t_3$
$t_2=7-6$
$t_2=1$

The value of $t_2$ is 1.