A sequence of numbers is a list of numbers supposedly in order (decreasing or increasing) and there may or may not be a pattern to the sequence, that is, there may or may not be a rule which applies to establish the next term in the sequence from what appears in the given terms.
You can have a sequence of geometrical shapes instead of a number. A typical example is a "snowflake" sequence. We start with an equilateral triangle, then on each side we take the central third of each side and take that to be the base of a smaller equilateral triangle. Build the triangle on each of the three sides of the original equilateral triangle and erase the base of these smaller triangles. The resultant shape will consist of a new set of 6 smaller equilateral triangles stuck together, surrounding a hexagon. This is the second shape in the sequence. The next shape is built by repeating the process, making 18 tinier equilateral triangles on the exposed sides. So we've created a (geometric) rule which establishes each succeeding shape. The shape more and more resembles a snowflake with successive generations.
These shapes constitute a sequence. If we didn't know the rule, we would see the sequence and try to work out what the pattern (the rule) is to generate the next shape. To make this more mathematical, we could count the number of sides and write these down in a sequence, corresponding to the sequence of shapes. Then we would need to find the rule which gives us successive terms (numbers of sides) in the sequence. This may not be as easy as spotting the geometric rule used for constructing the shapes.
Another example is to start with a rule which does not generate a sequence of numbers in ascending or descending order but generates a set of values which may ascend, descend, or generally wiggle around. Nevertheless, there is pattern, which is the rule we started with. But if a sequence is an ordered set of numbers, we don't have an ordered set.
Sometimes it's easy to see a pattern in a sequence but not so easy (or maybe not possible) to be able to write a formula which gives us a particular term in the series. We would simply have to plod on using given terms to find successive terms using the pattern we identified, until we reach the particular term in the series. For example: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... where the 3rd term is the sum of terms 1 and 2; the 4th term is the sum of terms 2 and 3, and so on (Fibonacci series or sequence).
One final example is: 6, 7, 9, 8, 6, 8, 6. This is the sequence, but what's the pattern? These are the numbers of letters in the days of the week (Monday, ..., Sunday). The pattern in this case is not really mathematical at all, and the sequence contains only 7 numbers. The pattern is based on English spelling!
Mathematicians and other scientists are always looking for patterns in all sorts of sequences. For example, the relative distance of the planets in the solar system from the sun. The distances can be measured, but is there a pattern relating them? Bode's Law seems to suggest there might be, but it's an empirical law (an approximation to a rule based on measurement, not a known physical law or mathematical rule). So you can have lots of sequences that look like there should be a pattern relating the terms of the sequence, and, despite strenuous efforts, you can't find a logical rule. But you keep searching...
