A fixed step between neighbours is the whole definition

0:00A chapter can open with a handful of situations and no question attached. Six of them. A salary that starts at eight thousand and goes up by five hundred every year. The rungs of a ladder, eight of them, measured from the bottom upwards: forty-five centimetres, forty-three, forty-one, and on down to thirty-one at the top. A savings scheme in which the amount becomes five quarters of itself every three years.

0:32The number of small squares inside a square of side one, then of side two, then of side three. A money box that gets a hundred on the first birthday and fifty more each year after that. And the pairs of rabbits in a colony, counted month by month: one, one, two, three, five, eight. Six lists of numbers. Three of them are the same kind of thing and three of them are not, and telling which is which is the whole of this video.

1:05Take each one and ask a single question of it. To get from any entry to the next, what do you do? The salary: add five hundred. Then add five hundred again. Every time, five hundred. The ladder: forty-three take away forty-five is minus two. And again, minus two. Seven times over, minus two. The money box: fifty, then fifty, then fifty. The savings scheme takes eight thousand to ten thousand, so two thousand was added. Ten thousand to twelve and a half thousand, so two thousand five hundred was added. Then three thousand one hundred and twenty-five.

1:50The amount being added is growing every time. The squares are one, four, nine, sixteen. Add three, then five, then seven, then nine. Growing as well. And the rabbits go up by nothing, then one, then one, then two, then three. Three lists with one unchanging step, and three without. Only the first three are going to get a name. An arithmetic progression is a list of numbers in which every entry after the first is the entry before it plus the same fixed number.

2:26That is the whole definition. Nothing about getting bigger. Nothing about a pattern you can see. One fixed number, added again and again. The numbers making up the list are called its terms. The one it starts from is the first term, and it is written a. The fixed number is called the common difference, and it is written d. Common, because every neighbouring pair in the list shares it.

2:54It does not say the list grows. It does not say d is positive. It does not even say d is anything other than nothing. Everything left in this video is that one sentence, pushed on. If you are handed a list and asked for its step, you subtract. But which one from which? Take six, three, nought, minus three. Subtract the smaller from the larger, the way you were taught to do with lengths, and you get six minus three, which is three.

3:27Now build the list from that. Start at six, add three, and you get nine. But the list in front of you says three. The rule is always the later term minus the earlier one. Three minus six is minus three, and that rebuilds the list exactly. That was checked on every one of six hundred and twenty-five lists in a sweep. Later minus earlier rebuilds all six hundred and twenty-five of them.

3:58The reverse reading rebuilds twenty-five — and those twenty-five are exactly the lists whose step is nothing, where the two subtractions are the same subtraction anyway. The sign is not decoration. It is the direction. So the step is allowed to be negative, and the ladder has already shown you one. Minus two, and the rungs shorten as you climb. It is allowed to be nothing as well. Three, three, three, three is an arithmetic progression with a common difference of nought. The list does not move, and the definition never asked it to.

4:40And a list can walk straight past zero without anything special happening. Minus three, minus two, minus one, nought steps by one the whole way, and the entry that happens to be nought is just an entry. Three pictures, one definition. Arrows of equal length pointing right. Arrows of equal length pointing left. And no arrows at all. Now the one that catches people. The savings scheme grows by a fixed factor — five quarters, every period, without exception. Fixed sounds like fixed. Is that not an arithmetic progression?

5:24It is not, and the reason is that a fixed factor and a fixed addend are different rules. Multiplying by five quarters adds a quarter of whatever you happen to have, and what you have keeps changing. This was swept rather than argued. Two hundred and eighty-nine lists were built by repeated multiplication, over a grid of starting values and a grid of factors. Thirty-three of them came out arithmetic. Two hundred and fifty-six did not.

5:57And the thirty-three are not a scattering. Every one of them either multiplies by one — which does nothing, and gives a list that never moves — or starts from nothing, where every entry is nought. Sixteen of them are of that second kind. Outside those two escapes, repeated multiplication never gives you a fixed step. How much do you have to know to have the whole list? Two numbers. The first term, and the common difference. Nothing else at all.

6:31From a equals six and d equals three you get six, nine, twelve, fifteen. From a equals six and d equals minus three you get six, three, nought, minus three. And neither number does the job on its own. In that same sweep, six hundred and twenty-five pairs give six hundred and twenty-five different lists — every pair its own list, no two alike. Throw the step away and keep only the first term, and the six hundred and twenty-five collapse to twenty-five. Throw the first term away and keep only the step, and they collapse to twenty-five again.

7:12Two numbers, and both of them are doing work. Write the list out without choosing any numbers at all. It starts at a. One step along, a plus d. Two steps, a plus two d. Three steps, a plus three d. Now look at the multiplier sitting on the d. It is not the position in the list. It is the number of steps you have taken to get there.

7:38a plus two d is the third term, because you started at the first and stepped twice. The misreading is the one that makes the next topic look arbitrary. Reading a plus k d as the k-th term was checked against the lists themselves at four thousand five hundred separate places. It fails on six hundred of the six hundred and twenty-five. The twenty-five it survives are the flat lists, where every term is a anyway.

8:11So underneath each entry, write the step count: nought, one, two, three. In practice you are handed the list, and you read the step off it. Ten of them, one after another, and the step falls out of each in one subtraction. One, two, three, four: step one. A hundred, seventy, forty, ten: step minus thirty. Minus three, minus two, minus one, nought: step one. Three, three, three, three: step nothing.

8:43Minus one, minus one and a half, minus two, minus two and a half: step minus a half. Heights running from a hundred and forty-seven up to a hundred and fifty-seven: step one. A week of minimum temperatures rising by a tenth of a degree a day: step a tenth. A loan balance falling by fifty a month: minus fifty. A row of prizes rising by fifty: fifty. Savings totals rising by fifty: fifty.

9:17Ten lists, ten steps. Four of the ten are not positive, and one of them is nothing at all. Here is one worth doing slowly, because it does three awkward things at once. Three halves, one half, minus a half, minus three halves. The first term is three halves. The step is one half minus three halves, which is minus one. Fractions. A negative step. And the list walks past zero between the second entry and the third.

9:50None of that is a special case. The step is minus one from the first pair to the last, and the list is an arithmetic progression the whole way down. Awkward to look at is not the same as different. One more thing falls out of the definition, and it is the definition read sideways. Take any three entries in a row and call them a, b and c. The step from the first to the second is the same as the step from the second to the third. That is what being an arithmetic progression means.

10:29So b minus a equals c minus b. Move the b's to one side and you get two b equals a plus c. The middle one is the average of its two neighbours. It is the same fact wearing different clothes, and it hands you a test you can run without subtracting anything. It was checked over two thousand one hundred and ninety-seven triples, of which eighty-five have equal steps. The average test and the equal-step test agreed on every single one — never once did one hold while the other failed.

11:06And beside it ran a mis-stated version: the middle IS the sum of the outer two, with the halving left off. That one parts company a hundred and eighty-six times. The disagreement is what makes the agreement worth having. A word on the checking, because a definition is the hardest thing in the subject to test. 'A list is arithmetic exactly when its differences are equal' is not a theorem. It is what the words mean. Read the differences, report that they are equal, and you have restated the sentence.

11:45Lists were BUILT by adding, and by adding only — no formula, no multiplication anywhere in the building. Then they were read back by taking the steps. Then tested a third time by the middle-of-any-three, which forms no difference at all. And beside those three, a fixed-factor reading, wrong on purpose, filed through the same line as the others. One thousand one hundred and sixty-four lists in all. Six hundred and fifty-eight arithmetic, five hundred and six not.

12:22The sweep also carries near misses on purpose: two hundred and fifty lists that are arithmetic except for one entry moved by one. A check that stops as soon as two steps agree accepts a hundred and twenty-five of those. All three honest routes reject all two hundred and fifty. That is what the near misses are there for. An arithmetic progression is a list whose step from each entry to the next is one unchanging number.

12:57The step is the later term minus the earlier one, sign included, and it is free to be negative, or to be nothing. A fixed factor is a different rule, and outside two escapes it never gives you one. Two numbers — the first term and the step — rebuild the entire list, and neither of them does it alone. In a plus k d, the k counts the steps taken, not the position reached.

13:25And the middle of any three entries is the average of its neighbours, which is the definition looked at from the side. One fixed number, added again and again. Everything else follows from that.