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.