Sometimes numbers, real or complex, are not enough to accurately describe a situation. Often, many numbers are needed.

This is particularly true if we wish to describe an evolving process, with outputs generated in a sequence over time. Examples include: number of cars produced on an assembly line, closing prices of a particular stock, and annual sunspot counts.

Any such example carries with it some sort of regular accounting. We count the number of cars off the assembly line each hour, we look at a stock's closing price each day, we tabulate our sunspot data each year. Regardless of the time interval, there is the underlying notion of a first value, a second value, a third value...

We model these situations using functions  f  with inputs that are counting numbers. (Either the positive integers 1, 2, 3, ... or the natural numbers 0, 1, 2, ...) Thus, f(1)  gives the first value in the sequence, f(2)  gives the second value, and so forth. The outputs may be real or complex, depending on what is being modeled, but a sequence always has inputs that provide us with a simple "count".

Typically, we write  a₁  for  f(1) ,  a₂  for  f(2) , etc. We can then display a sequence in the familiar form of  a₁ , a₂ , a₃ , ...

We call the rule for the function  f  the general term of the sequence. For example, the general term:

a_n = f(n) = 1/(n + 1)

would describe the sequence:

1/2, 1/3, 1/4, ...

and the general term:

b_n = g(n) = (–1)ⁿ/(n + 1)

would describe the sequence:

–1/2, 1/3, –1/4, 1/5, –1/6, ...

Notice that these two examples implicitly assume that we are counting with the positive integers (beginning with  1). If we choose instead to count with the natural numbers (beginning with  0), then the two general terms describe different sequences:

a_n  describes  1, 1/2, 1/3, ...

and

b_n  describes  1, –1/2, 1/3, –1/4, 1/5, ...

The choice of how we will count depends on the situation. For example, we would not wish to begin at  0  with a general term of  1/n .

If the starting point of a sequence  f(n) = a_n  is understood, we often use the shorthand  {a_n}  for the sequence itself. If the starting point needs to be given more explicitly, we write things like:

where  P  and  N  represent the positive integers and natural numbers, respectively.

When describing sequences, we must be very clear about where we begin. The general term describes everything that comes after...