A rule that turns a position number into a term

0:00Two perfectly ordinary questions, and each of them spills out a list of numbers. How many ancestors do you have, going back three centuries, if you count a generation as thirty years? Three hundred divided by thirty is ten generations, so there are ten answers to give. Two parents, four grandparents, eight behind them, and at the tenth generation, one thousand and twenty-four. That list stops, and it stops because the question stopped it.

0:33Now divide ten by three, and write down what you have in your hand at each stage of the division. Three, then three point three, then three point three three, and that one never finishes. Both are collections in a definite order, and one of them runs out while the other does not. What makes them the same kind of object is not the numbers you can see in them.

0:59It is that each has a first entry, a second, a third, and no doubt anywhere about which is which. Because the order is the whole point, the positions get names of their own. The entry in first place is a one, the entry in second place is a two, and the entry in place n is a n. That small number written below is not part of the value.

1:26It is the question being asked, and the term is the answer that comes back. Keep them separate, because almost every mistake here is those two tracks touching. The upper track is a strip of positions: one, two, three, and onward. The lower track is a strip of terms, and something carries you from a position on the upper track down to a term on the lower one. That something is the whole content of the idea.

1:56The list of numbers is only what it says when you ask about the first few positions. Erase the answers and the object is still there; erase the rule and there is nothing left at all. So a fair question is where a list stops, and the answer is never how many terms you happen to have seen. Take the ancestor rule: two raised to the position. Take the doubling rule: two raised to the position, on and on with no end.

2:28Those are the same rule. Asked out to position twenty, they disagree at not a single place they both have. The only difference between them is that the second one has ten positions out there that the first one does not. One of them was handed the numbers one to ten and the other was handed the counting numbers. Change nothing but the set of positions, and a list that runs out becomes a list that does not.

2:58That is why the verdict belongs to the positions and to nothing else. Ask how far the questions go, not how many answers you were given. There is a second thing never-ending does not mean. It does not mean the values run away. Go back to dividing ten by three and stopping at each stage. Every one of those terms sits below ten thirds, because every one of them is that division cut short.

3:28Take the first forty positions, and the number of terms that reach ten thirds is nought. The fortieth is short of it by one part in three followed by thirty-nine noughts, and it is still short. So here is a list with no last entry whose entries never get anywhere. Beside it, put two raised to the position. That one passes ten thirds at position two, and of its first forty terms exactly one stays underneath.

3:59Never-ending is a statement about how many questions there are, not about how large the answers get. Most rules arrive the other way round: you are shown some terms and asked for the rule. Two, four, six, eight. Two is twice one, four is twice two, six is twice three, eight is twice four, so the term at position n is two n. That reasoning is not finished, and it is worth seeing exactly why.

4:31Here are seven candidate rules for that same opening. Six of the seven agree with all four terms you were shown. They are built that way: each is two n plus something that comes to nothing at the first four positions and to something everywhere else. One of the six says thirty-four at position five, where twice five is ten. So test the rule where it was not read from: at position twenty-three it must say forty-six, and at twenty-four, forty-eight.

5:04Of the six that fitted, exactly one is left standing, and that is the only reason to believe it. A rule does not have to be an expression in the position at all. It is allowed to lean on the terms already built. Start with two terms, both of them one. From then on, each term is the total of the two immediately before it. One and one is two, one and two is three, two and three is five, three and five is eight.

5:36Notice the opening: two terms with the same value, sitting in different places. That is not a fault, and it is the quickest way to see that a sequence is about positions, not values. Six terms in, there are only five different numbers among them. Divide each term by the one before it, and you get one, then two, then three halves, then five thirds, then eight fifths. The rule is complete, it is unambiguous, and there is no expression in n anywhere in it.

6:12The two kinds of rule feel similar, and the difference between them has a price you can count. Take the rule that multiplies together one less than the position, the position taken from two, and three more than the position. Ask it for position twenty. The three brackets are nineteen, minus eighteen, and twenty-three, and their product is minus seven thousand, eight hundred and sixty-six. The number of earlier terms you needed to get there is nought.

6:45Now take a rule that starts at one and adds two to the term before. Ask that one for position twenty, and the answer is thirty-nine, but you cannot have it until you have built nineteen earlier terms. Not because it is harder arithmetic; because the rule genuinely does not say anything about position twenty until position nineteen exists. There is a nice refinement here too: a rule with two starting terms that only looks one step back never consults the first of them.

7:21Reaching its fifth term costs three earlier terms rather than four, and only walking it tells you that. Now the case that forces the definition to be careful. Two, three, five, seven, and onward: the primes, in order. There is a first one, a second one, a twenty-fifth one, which is ninety-seven, and no ambiguity anywhere. But nobody hands you an expression that turns the position into the prime standing there.

7:53Here are four expressions someone might offer for it. Scored against the first twelve primes, they get nine, twelve, eleven and ten of the twelve wrong. The best of them still misses nine positions out of twelve. That does not prove no expression exists, and it is worth being exact about the difference. It shows that these four fail, and that a description in words is doing the work here instead.

8:22The demand was never that you can write the rule as a formula; it was only that each position determines its term. So gather the three cases. An expression in the position, a rule that leans on earlier terms, and a description in words with no expression at all. In every case there is a set of positions, and every position in it determines exactly one number. That is a function whose inputs are the counting numbers, or a front stretch of them when the list runs out.

8:59The word exactly is carrying real weight there. Take a relation that sends four to two and also sends four to minus two. Both are square roots of four, the relation is perfectly well defined, and it is not a sequence. At position four it offers two numbers, so it never says what the fourth term is, and it has to be turned away rather than quietly tidied up. Compare a rule that sends one to five and two to five as well: one number at each position, so that is fine.

9:35Repeated values are allowed; a position with two answers is not. Two habits of thought are worth breaking now. The first is that a term and its position are somehow the same thing. Take the seven sequences that have appeared so far, and ask each of them about its first twelve positions. That is eighty-four pairs of a position and a term, and the two are equal at exactly one of them.

10:05The odd numbers put a one in first place, and that coincidence proves nothing. The second habit is expecting the terms to climb. Take three less than the position, all over four. Its first three terms are minus a half, minus a quarter, and nought. Two of them are below nothing and one of them is nothing, and it is a perfectly good sequence. Order in a sequence is order among the positions, and the values are free to do whatever the rule makes them do.

10:40One more trap, and it is the one that costs marks. Rules that switch sign do it with minus one raised to a power, and the power is usually one less than the position. Minus one to the power n minus one, times five to the power n plus one. At position one the exponent is nought, so that first term is positive. Twenty-five, then minus one hundred and twenty-five, then six hundred and twenty-five, changing sign four times across the first five terms.

11:15Read the exponent as n instead of n minus one and every one of those five signs comes out wrong. Not some of them; all five, because the whole pattern is shifted by one place. The same trap sits in minus one to the n minus one times the position cubed, which is positive seven hundred and twenty-nine at position nine, not negative. And the same off-by-one is why the first position is one rather than nought.

11:47Start the even numbers at position nought and you get nought, two, four, six, which agrees with two, four, six, eight at not a single place. Fix that convention now, because every rule after this carries an exponent of n minus one because of it. So the object is the assignment, and the numbers are what it says when you ask it. When a list of numbers is put in front of you, three questions are worth asking.

12:18How far do the positions go, because that and nothing else settles whether the list runs out. What carries a position to its term: an expression you can substitute into, a rule that leans backwards, or a description you can only follow. And does each position get exactly one answer, because that is the whole of the definition. A sequence with no formula is not a lesser sequence. And a term is never its position, however often the two happen to coincide.

12:53Everything after this is the same object with more asked of it: what the terms add up to, and what they add up to when there are infinitely many.