Concept

The Fibonacci Sequence

The Fibonacci sequence is a well-known sequence, where the first two terms are both

1 .

Subsequent terms in the sequence are generated by adding the two preceding numbers.

Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...

Starting from the third term onward, the sequence can be represented using a rule.

a_{1} = 1 a_{2} = 1 a_{n} = a_{n - 1} + a_{n - 2}

This is an example of recursive rule as the

n th

term depends on the values of the previous two terms. This indicates that to find, for instance, the seventh term

a_{7}

of the sequence, it is necessary to have information about the fifth and sixth terms.

	$a_{n} = a_{n - 1} + a_{n - 2}$
$n = 7$	$a_{7} = a_{6} + a_{5}$
$a_{6} = 8$ and $a_{5} = 5$	$a_{7} = 8 + 5$
Evaluate	$a_{7} = 13$

The sequence is named after the Italian mathematician Leonardo Fibonacci, who used it to describe how pairs of rabbits increases as they multiply under certain conditions. Sometimes, the first two numbers are defined as $a_{1} = 0$ and $a_{2} = 1 .$ This, however, does not change the sequence in any way other than by increasing the index of every term by one compared to the previously mentioned definition.

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