Sequence is a big word. It can sound scary, like an impending thunderstorm heading your direction, approaching with thunderous laughs. Sometimes we could swear we saw a set of evil eyes looking ominously at us from the clouds.
Don't worry. A sequence is just a list of numbers. For example, pick an integer and right it down. Now pick two more and write them down behind the first. Separate each number with a comma. You just made a sequence. Nothing scary about that, right? The individual numbers you just wrote are called terms. The sequence you just made up should have three terms. Because it has three terms–the list of numbers ends–it's finite.
Let's make it a little more complicated. Get out your writing hand of non-stop, legible scribble and your infinitely long #2 pencil. No, this is not a never-ending standardized exam. Start writing down numbers, separated by a comma, and never stop. That sequence is infinite.
Be Careful: In math, sequences and series are not the same thing.
We can write down a finite sequence by writing down its terms with commas in between. If a sequence is infinite, we don't need a special hand or #2 pencil. Instead, we write down some of its terms, and then we write some dots (...) to indicate that the sequence continues forever in the same pattern.
The following are finite sequences:
1, 2, 3
1, 1, 1, 1, 1, 1, 1, 1
The following are infinite sequences:
1, 1, 1,...
1, 2, 3,....
While we might only encounter infinite sequences in calculus class, nothing about the definition says sequences have to be infinite. Finite sequences are okay, too.