Fibonacci Sequences as Second Order Difference Equations

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.