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:

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)

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

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

This is correct.

This is correct.

This is correct.

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

b)

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

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

This is correct.

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:

  given and

Example 2

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

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

and

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

Example 3

Find the value of given and .

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

Now we know  we can work backwards to find .

The value of  is 1.