The Successor Formula

Suppose we know a term in the Fibonacci series and we want to know the term that directly follows it. How would we go about doing this? Without knowing the term that precedes the term we know, we can't use the definition of the series. We could use Binet's formula, but that takes a lot of work. What if there were an easier way to calculate a term of the series knowing only the term that precedes it?

There is, of course, such a way, known as the Successor Formula, as it finds the successor to each term. This formula uses a function known as the "greatest integer" function. The greatest integer function is denoted [ x ], and it is defined as the greatest integer less than or equal to x. For example:

[ 4.2 ] = 4
[ 4 ] = 4

The Successor Formula takes the argument of any term of the Fibonacci series and is of the form:

Note that the value of x in this equation is not the numerical order of the Fibonacci number, it is the Fibonacci number itself. For instance, if you put 3 into the Successor Formula, you will not get the 4th Fibonacci number, you will get the Fibonacci number that comes after 3 in the series.