The General Term of Fibonacci Sequence

In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8, … has no visible pattern, but the sequence is generated by the recurrence relation given by

a1=a2=1
a3=a1+a2
an=a(n-2)+a(n-1), n>2

This sequence is called Fibonacci sequence.

In some cases it is not easy, or even possible, to give an explicit formula for an. In such cases, it may be possible to determine a particular term in the sequence in terms of some of the preceding terms. This relationship is often referred to as a recurrence. For exam- ple, the sequence of positive odd numbers may be defined by

a1=1 and a(n+1)=an+2, for n≥1.

The initial term is a1=1, and the recurrence tells us that we need to add two to each term to obtain the next term.

A pair of rabbits are too young to produce in their first month. In the second, and every subsequent month, they produce a new pair. Each new pair of rabbits produce a new pair in their second month and in every subsequent month (see the figure below). Assuming no rabbit dies, the number of pairs of rabbits at the start of the 1st, 2nd, 3rd, …, 6th month, respectively are:

1,1,2,3,5,8
 
fibonacci sequence

The Fibonacci sequence comprises the numbers

1,1,2,3,5,8,13,21,34,…

where each term is the sum of the two preceding terms. This can be described by setting a1=a2=1 and a(n+2)=a(n+1)+an, for n≥1.

The general term of a sequence can sometimes be found by ‘pattern matching’.

In general, however, finding a formula for the general term of a sequence can be difficult. Consider, for example, the Fibonacci sequence:

1,1,2,3,5,8,13,21,34,….

We will discuss in the History and applications section how to show that the nth term of the Fibonacci sequence is given by
the explicit formula of fibonacci sequence

(The general term of Fibonacci sequence)

This is a very surprising result! (It is not even obvious that this formula will give an integer result for each n.) You might like to check that this formula works for n=1, 2,3.
General Term, Explicit rule, explicit formula and functional definition have the same meaning.

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