Patterns, sequences, and rules

In brief

Look at a row of growing squares built from matchsticks: one square takes four sticks, two joined squares take seven, three joined squares take ten. Ask what comes next and most people answer thirteen without hesitation, and they are right. But the interesting question was never "what comes next." It was "what rule produces every term in this list, and how do you know it is the rule, rather than merely a rule that happens to agree with the terms you have seen so far." A short list of numbers never proves a single generating rule; it only rules out the rules that disagree with it. Learning to state the rule precisely, and to notice when the evidence for it is thinner than it feels, is the actual skill behind spotting a pattern.

The full treatment

First look: the matchstick squares

Build the squares one at a time and count sticks as you go: four for the first square alone, seven once a second square is attached sharing one side, ten once a third is attached, fourteen sticks would be wrong, it is thirteen, since each new square after the first only needs three new sticks, borrowing one side from its neighbor. Lay the counts in a row: four, seven, ten, thirteen. The regularity jumps out, each term is three more than the last, but noticing that regularity is not yet the same achievement as stating a rule that tells you the fiftieth term without building forty nine squares of matchsticks first.

Building the idea: recognizing regularity versus stating a rule

Separate two activities that feel like one. Noticing repetition is comparing consecutive terms and seeing that something stays constant, here the difference of three sticks between neighbors. Stating a generative rule is writing a precise recipe that produces the value at any position without needing every earlier value listed out. Two kinds of generative rule serve this purpose. A recursive rule gives a starting value and a step connecting each term to the one before it: here, the first term is four, and each later term equals the previous term plus three. A closed form, or explicit rule, gives the value directly as a function of the position alone, with no reference to earlier terms: here, the term at position n equals three n plus one. Both rules generate the identical list of numbers; they differ in what they require to compute a distant term.

The formal model: sequences as functions from position to value

A sequence is a function whose domain is the positive whole numbers, one, two, three, and so on, and whose output at position n is written a sub n. The recursive rule for the matchstick sequence is stated as a sub one equals four, and a sub n equals a sub n minus one, plus three, for every n greater than one. The closed form is a sub n equals three n plus one. Check the closed form against the recursive one at a few positions: at n equals one, three times one plus one is four, matching the stated start; at n equals two, three times two plus one is seven, matching seven sticks for two joined squares; at n equals four, three times four plus one is thirteen, matching the count reached by hand. The recursive rule needs every earlier term to reach a distant one; the closed form lets you compute the term at position one hundred directly, without building ninety nine intermediate squares.

Derivation: from pattern to closed form, and why the closed form can be trusted

The method of finite differences turns a noticed regularity into a closed form. Compute the difference between each consecutive pair of terms: seven minus four is three, ten minus seven is three, thirteen minus ten is three. A constant difference of three signals that the sequence grows by the same fixed amount at every step, which is the defining feature of what is called an arithmetic sequence, and any arithmetic sequence with first term a sub one and constant difference d has closed form a sub n equals a sub one, plus d times the quantity n minus one. Substituting a sub one equals four and d equals three gives a sub n equals four plus three times the quantity n minus one, which simplifies to three n plus one, matching what was found by inspection.

Why trust this beyond the four terms actually computed. Because the recursive definition states, by its own construction, that every new term is obtained from the last by adding exactly three, with no exception built in. Unwinding that definition n minus one times from the starting value a sub one produces exactly a sub one plus d times the quantity n minus one, for any n whatsoever, not merely the four checked by hand. This unwinding argument is an informal version of a proof technique called mathematical induction, treated fully once proof itself becomes the object of study; here it is enough to see that the justification comes from the mechanism generating the sequence, the fixed rule of growth by three per new square, and not merely from the four numbers that happened to be listed.

Why a finite list never proves a rule

A caution belongs alongside every success here. The list two, four, six, eight is compatible with the obvious rule a sub n equals two n, but it is also compatible with a rule that matches these four terms exactly and then produces sixteen next, or any other value you like; a technique called polynomial interpolation can always construct some formula agreeing with any finite list of numbers and then departing from it in whatever way is specified afterward. A handful of terms is never logically sufficient to force a unique continuing rule; it is only sufficient to eliminate the rules that disagree with those particular terms. What justifies preferring a sub n equals two n over the engineered alternative is not the four matching terms, which both rules share, but an independent reason for the mechanism, such as counting even numbers in order, that makes the simple rule the one actually operating.

Lineage

The figurate numbers, triangular numbers, square numbers, and pentagonal numbers, studied by the Pythagoreans in the fifth century before the common era, are among the earliest recorded examples of sequences generated by an explicit geometric growth rule, arranging dots or pebbles into growing shapes and reading off the counts. Fibonacci's Liber Abaci, written in 1202, introduced to a wide European readership a sequence defined recursively, each term the sum of the two before it, starting from a problem about the growth of a rabbit population, an early and widely known recursive rule outside its Sanskrit and Arabic mathematical origins where similar recurrences had already been studied. The general theory of sequences as functions defined on the positive integers matured alongside the function concept in the eighteenth and nineteenth centuries. The caution that finite data never uniquely forces a continuing rule was sharpened into a general philosophical problem by Nelson Goodman in the twentieth century, who showed that any finite set of observations is equally compatible with wildly different hypotheses about what comes next, a point directly relevant to trusting a spotted numerical pattern.

The strongest case for it

A rule found this way, whether recursive or closed form, predicts every future term without recomputing the entire history, lets you compare how quickly different processes grow, and is the earliest concrete instance of the general function concept. Recurrence relations built from exactly this kind of rule finding model an enormous range of real processes: compound interest growing a balance by a fixed rule each period, a population changing by a fixed proportion each generation, and the running time of an algorithm that calls itself on a smaller version of its input. The distinction between recursive and closed forms matters practically as well as conceptually, since a closed form can be evaluated for a distant term in a single step where a recursive rule would require computing every intermediate term first.

The strongest case against it

Pattern spotting from a short list is a hypothesis, never a proof, and puzzles are frequently constructed specifically to exploit an unjustified leap from a few matching terms to a false general rule. A first common misconception is believing that listing more terms always removes the ambiguity; it reduces the number of surviving candidate rules in practice but never eliminates every possible alternative rule in principle, since an engineered rule can always be built to match any additional finite stretch and diverge later. A second common misconception is treating recursive and closed forms as interchangeable in cost as well as in output; a recursive rule can be far more expensive to evaluate at a distant position than a closed form, and not every sequence defined recursively has a closed form that can be written down at all, so the existence of one description does not guarantee the existence of the other.

Where it stands now

The mathematics of generative rules, recursive definitions, closed forms, and the method of finite differences connecting patterned data to an explicit formula, is settled and uncontested. The pedagogical emphasis on justifying a rule rather than merely guessing it from a handful of terms reflects a long standing consensus in mathematical practice, tightly connected to the concept of mathematical induction and proof, which gives the justification its full rigorous form.

Test yourself

Consider the sequence one, two, four, seven, eleven, continuing by a rule you must discover, without using the words "difference" or "increase" until you have found and stated both a recursive rule and a closed form for the term at position n. Verify your closed form against all five given terms. Then construct a second, different rule that also produces exactly these five terms but departs from your first rule at the sixth term, stating explicitly what the sixth term is under each rule, to demonstrate concretely why five terms alone could never have forced your original answer to be the only possible one.