What counts as a sequence, and how would we know that it isn't deceiving?

Question

A sequence, in my understanding is just a list of numbers in order. In that case, if I write down random numbers with no pattern at all except for the fact that it gets larger, is it a viable sequence?

In that case, given a graph, how can we tell if a sequence is linear? Everywhere I've looked just says that linear sequences lie in a line. However, what if the sequence graphed is infinite and is linear for some, but then curves? In that case, couldn't there be sequences that go "1,2,3,4,9,54,321,556..." where if the first few numbers were graphed, we would think it's linear.

So, my point is, how can we be sure if a sequence is linear other than "the points lie in a line." And what is even considered to be a sequence? As an extra note, when do we ever know the rule of a sequence if there is no context? Can't a sequence always be deceiving and we can never be sure of its nature? Finally, am I overthinking this or does nobody talk or teach this ever and why?

Answer 1

I'm sorry for how long this answer is. But your question is a deep one and I wanted to take it seriously.

Mathematics operates very differently from empirical sciences like physics and chemistry. Sciences take finite collections of observations and try to guess the general law that produced them. For example, based on what I know about chemical elements, can I predict the properties of element 118? Will it be inert? Will it be radioactive? Then I answer these questions by getting hold of an atom of element 118 and observing its actual behavior. This is what seems to be behind your question about how we can be sure that a sequence is linear.

But mathematics mostly doesn't work this way. An atom of element 118 is a real object with observable properties in the real world. A sequence is nothing at all like this. We cannot use any scientific instrument like a Geiger counter to find out if a sequence is linear.

Mathematics has a completely different approach. Mathematical objects are abstract, defined by their abstract properties.

The goal of mathematics is to understand the relationship between different abstract properties.

This is why mathematics is able to deal with infinite objects such as infinite sequences, infinite sets like the set of odd integers, or the infinite lines and planes of geometry. These objects can't be represented explicitly. Instead they are handled abstractly and descriptively. We can make a mark on a piece of paper to help us think about lines in the plane, but papers are not mathematical planes and marks on papers are not mathematical lines. Mathematical objects are not drawings, they are descriptions of properties.

Suppose someone hands us a list of numbers and asks “Is this sequence linear?”:

$$ 1, 3, 5, 7, 9, \dots $$

We can't really answer, because this is not a sequence. It's just a list of numbers with three dots written at the end.

We might guess that this person meant to ask if the sequence of positive odd integers was linear. But that is not what they did ask, and we don't know if that was what they meant. Mathematicians hate questions that begin “find the next number in this list”. To a mathematician, a finite list is not a sequence, and any number, or none at all, could be next.

Instead, the mathematical version of the question looks like this:

Is the sequence of positive odd integers a linear sequence?

How do we resolve this? Not by scientifically examining the sequence itself. It is infinite and also has no concrete existence. Instead, we ask what properties the abstract sequence has. One way it could be defined is:

The sequence of positive odd integers, called $\mathcal O$, is the sequence whose $n$th term is $2n-1$.

(Just for this post, let's agree that this needs no further explanation.)

And next, what does it mean for a sequence to be linear? One way is:

A sequence is linear if there exist numbers $a$ and $b$ for which the $n$th term of the sequence is always exactly $an+b$.

With these definitions, it's trivial to conclude that $\mathcal O$ is linear. If we take $a=2$ and $b=-1$ in the definition of ‘linear’, it matches up exactly with the definition of $\mathcal O$. Even though $\mathcal O$ is an infinite object, we can conclude that it is linear by considering its properties and how they relate to the property of being linear.

Here's a less silly example. Let $S$ be the sequence whose $n$th term is the difference between the $n$th perfect square and the previous square. So for example the 10th element of $S$ is the difference $$10^2 - 9^2 = 19.$$ Is sequence $S$ linear?

$S$ here is completely specified, much more clearly than if I were to list the first million terms. Many sequences begin with those same terms. You asked “when do we ever know the rule of a sequence if there is no context?” This is right on the nose! We cannot know the rule of a sequence from a list of some of its terms, we can only guess at what the writer might have had in mind.

But the abstract description says literally everything there is to know about $S$, not just the first million terms but all the rest too. What's the one-millionth term of $S$? It's $1000000^2 - 999999^2 = 1999999$. What's the term after that? It's $1000001^2 - 1000000^2 = 2000001$. We don't need too see a list of terms, because a sequence is not a list of terms, it's a collection of properties.

How can we decide if $S$ is linear? Not by dealing with its individual terms, but by dealing with its properties. We can use simple algebra to show that $S$ is actually the sequence of positive odd integers $\mathcal O$, disguised:

1. $S$ is the sequence whose $n$th term is the difference between the $n$th perfect square and the previous square. (Because I said so.)
2. That is, the $n$th term of $S$ is $n^2 - (n-1)^2$
3. Algebra tells us that $n^2 - (n-1)^2 = n^2 - (n^2-2n+1) = 2n-1$
4. So the $n$th term of $S$ is equal to $2n-1$
5. That's exactly the property that defines the sequence of positive odd integers

We already knew that $\mathcal O$ was linear, and now we know that $S$ is equal to $\mathcal O$, so $S$ is linear. And notice: we can come to this conclusion without knowing even a single one of the terms of $S$.

That is how mathematics deals with infinite objects like sequences, and usually with finite objects too. Not by examining data, but by reasoning about properties.

Answer 2

What counts as a sequence?

A sequence of numbers is, simply put, a function whose domain is the natural numbers and whose range is the real numbers.

So, like any function, what counts as a function can be specified in various ways, but the most basic way is a formula.

For example, you ask How can we tell if a sequence is linear? The geometric answer is that the graph of the sequence lies on a line. But we know how to graph lines. We learn in precalculus that the graph of a (non-vertical) line can always be described as the graph of a function of the form $$f(t) = mt + b $$ where $m$ and $b$ are constants that must be given.

Replacing the input variable $t$ by a natural number variable $n$, and replacing function notation $f(n)$ by sequence notation $x_n$, we arrive at the formula $$x_n = mn + b $$ You gave the example 1,2,3,4,9,54,... Writing out these values in sequence notation we have $$x_1 = 1, \quad x_2 = 2, \quad x_3 = 3, \quad x_4 = 4, \quad x_5 = 9 $$ You can now use the formula $x_n = mn+b$ to prove that this sequence is not linear, like this. From just the first two values $x_1=1$ and $x_2=2$ you can determine that if this sequence is linear, in other words if it has the form $x_n=mn+b$, then the values of $m$ and $b$ must be $m=1$, $b=0$. The formula for the sequence must therefore by $$x_n=1n+0 \quad\text{or, more simply,}\quad x_n = n $$ But that is contradicted by the fifth term $x_5=9$. Therefore this sequence is not linear.

You can repeat this logic to rule out linearity of many sequences: any two terms of the sequence can be used to compute what $m$ and $b$ must be; and then if there's just one more term of the sequence that violates the formula $x_n = mn+b$, voila! You have proved that the sequence is not linear.

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More generally, sequences can be given by function formulas, or by recursion formulas, or other methods. The key here is that you need some method which rigorously defines $x_n$ for all values of $n$.

If someone hands you a list of the first $8$ terms of a sequence and gives you no further information, you can justifiably say that the sequence has not been defined fully.

On the other hand, if someone hands you the first $8$ terms and tells you one additional piece of information --- namely This sequence is linear --- then you can use any two of the terms to derive the formula for the sequence; you can test the formula on the $8$ given terms; if one of them violates the formula you can say to them No, it's not linear; whereas if all $8$ given terms do line up then you can use the formula to compute $x_9$ or $x_{10}$ or $x_{23805417}$ or any term you like.

Answer 3

I suspect the issue here is questions of the type "a sequence starts $1,2,4,8,16$, what is the next number?". For a while in your mathematical life, you think, fine, it's just doubling, then you come across this https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas and suddenly you get existential crises like these.

The fact is, when you're doing maths, that type of question - with someone who'll tell you you're wrong if a valid reason gives you a different number - doesn't come up. Not in that form, anyway. More usually, you will have some rule (like the circle cutting above) that lets you generate the first few terms by hand, but then you want to know what the $n^\text{th}$ term is.

The skill you get from doing "what comes next?" questions is being able to recognise common sequences. These allow you to conjecture a general rule, or maybe a few possible rules; you usually calculate as many terms as you can, or as many as you need, to rule out some possibilities. It's - generally - best to try sequences with simpler rules first (essentially, applying Occam's razor); that's where this skill is useful.

Once you have a sensible conjecture (that matches with data), it's then a case of proving it; that's how you know whether or not it's "right".

Answer 4

Say that you are given a computation black box $B$, i.e. a function that given an input $n$ return a number $B(n)$ which is the $n$-th element of the sequence. But you don't know anything else about $B$. Then you cannot tell if $B(n)$ is a linear function of $n$ since there is no way for you to try all values of $n$ in your finite lifetime. You need to gather some extra information about $B$ in order for you to establish linearity. But if you have no information about $B$ then there is really nothing you can say other than statements such as "I tried $B$ and it is linear up to the first billion elements".

Answer 5

A more precise definition of a linear sequence, rather than "the points lie on a line", is that the difference between successive terms is the same for all pairs of successive terms in the sequence (note that a one-term sequence always qualifies as a linear sequence because it contains no pairs of successive terms).

If you only know part of a sequence then all you can say is whether or not the part that you know is linear - you can say nothing at all about the rest of the sequence.

So if you know that the first four terms of a sequence are $1,2,3,4$ then all you can say is "the first four terms of the sequence form a linear progression". The next term could be $5$ or it could be $-75179386$; you have no way of knowing.