Building the rule for the nth term out of repeated addition

0:00Someone starts a job on eight thousand a month, and every year the salary goes up by five hundred. What is the salary in the twenty-fifth year? You know everything you need to answer this. The list starts at eight thousand and the step is five hundred, and that is all an arithmetic progression is. So you could just write it out. Eight thousand, eight thousand five hundred, nine thousand, and keep going until you have twenty-five numbers.

0:32That works, and it is not wrong, and nobody wants to do it. What we want is to arrive at the twenty-fifth entry without walking through the twenty-four before it. And the thing that stands between us and that is not a formula. It is a counting question, and it has a trap in it. Let us do it the slow way first, for the first five years, and watch what we are actually doing.

1:00Year one: eight thousand. That is where we start. We have not moved yet. Year two: add five hundred. Eight thousand five hundred. Year three: add five hundred again. Nine thousand. Year four: nine thousand five hundred. Year five: ten thousand. Now count the additions rather than the years. To reach year one we performed no additions at all. To reach year two, one. Year three, two. Year four, three. Year five, four.

1:34The first entry is where you stand before you move, so it costs nothing. Every entry after it costs exactly one step. That sentence is the whole topic, and everything that follows is bookkeeping on it. Look at the third year written out in full: eight thousand, plus five hundred, plus five hundred. Two identical additions in a row. And two identical additions in a row are a multiplication. Eight thousand plus two times five hundred. Nine thousand, the same answer.

2:12Now write that two in a way that remembers where it came from: it is three minus one. Eight thousand plus, in brackets, three minus one, times five hundred. Four ways of writing the same number, and the last one is the only one that mentions the year. The fourth year is eight thousand plus four minus one times five hundred, and the fifth is eight thousand plus five minus one times five hundred.

2:42The pattern is not subtle. Every multiplier is one below the year it delivers. Here is why, and it is worth being slow about, because this is where the mistake lives. Put the years along the top: one, two, three, four, five, six. Put the number of increments paid along the bottom: nought, one, two, three, four, five. Two rows of numbers, offset by exactly one, all the way along.

3:12The top row is a position. The bottom row is a count of steps taken. They are not the same thing and they never were, and the whole of this rule is the gap between them. The multiplier in front of the step is the bottom row, not the top row. It counts how far you have walked, not where you are standing. Say it as a sentence: to reach the nth entry you take n minus one steps.

3:42Try it on the fifteenth year, and let us do it honestly first. The fourteenth year, plus one more five hundred. And the fourteenth year is eight thousand plus thirteen copies of five hundred. Thirteen. Not fourteen. Because the first year is where we started and cost nothing. So the fifteenth year is eight thousand, plus thirteen five hundreds, plus one more. Thirteen and one is fourteen, and there is the number one below the year.

4:14Eight thousand plus fourteen times five hundred. Fourteen times five hundred is seven thousand. Fifteen thousand. Notice we never wrote out years two through thirteen. We only counted them. And now the question we opened with. The twenty-fifth year. Twenty-five minus one is twenty-four. Eight thousand plus twenty-four times five hundred. Twenty-four times five hundred is twelve thousand. Eight thousand plus twelve thousand is twenty thousand. Twenty thousand a month in the twenty-fifth year, and we wrote three lines.

4:55The long way would also have given twenty thousand. The long way is correct; it is just slow, and it is how the short way was found. Do not let anyone tell you the formula replaced the addition. The formula IS the addition, counted properly. Now strip the numbers out, because the argument never depended on them. Call the first entry a and the step d. The first entry is a. No steps taken, so no copies of d.

5:29The second entry is a plus d. One step, one copy. The third is a plus two d. The fourth, a plus three d. Write those multipliers in a column and look down it. Nought, one, two, three. Beside them write the positions: one, two, three, four. Every multiplier is its position minus one, and it is minus one because the first position paid nothing. So the entry in position n is a plus, in brackets, n minus one, times d.

6:04That expression is not a thing to memorise. It is an instruction, and it reads left to right. Stand at a. Take n minus one steps, each of size d. You are now on the nth entry. Stand at eight thousand. Take twenty-four steps of five hundred. You are in the twenty-fifth year. The step can be negative and the instruction does not change: you take n minus one steps in the other direction.

6:35Ten, seven, four has a step of minus three, so its twentieth entry is ten plus nineteen lots of minus three, which is minus forty-seven. And the step can be a fraction. Three halves, one half, minus one half has a step of minus one, and its twelfth entry is minus nineteen halves. One instruction, three quite different-looking lists, and no special cases anywhere. Now the mistake, because it is the most common error anywhere in this subject and it deserves a proper look.

7:14The mistake is to write a plus n d. It is not a silly mistake. It looks right. The nth entry, n copies of the step. Watch what it actually does. Ask it for the fifth year. Eight thousand plus five times five hundred is ten thousand five hundred. The fifth year is ten thousand. So ten thousand five hundred is wrong. But look at what it is. Ten thousand five hundred is the sixth year's salary. Exactly the sixth.

7:47Ask it for the third year and it hands you nine thousand five hundred, which is the fourth. Ask it for the tenth and it hands you thirteen thousand, which is the eleventh. It is not producing wrong numbers. It is producing the right numbers for the wrong position, every single time, one place too far along. That is why the salary is the right way to see it. The error is a wrong YEAR, and a wrong year is something you can feel.

8:20There is one place where the mistake costs nothing, and it is worth knowing about. If the step is nought, the list never moves. Three, three, three, three. Then a plus n d and a plus n minus one d are the same number, because both of them add nothing. So on a list that stands still the error is invisible. Every other list gives it away. I swept a hundred and sixty-nine progressions across the first thirty positions, five thousand and seventy positions in all.

8:55The rule was right every time. The mistake was right three hundred and ninety times, and all three hundred and ninety were on lists with a step of nothing. The other four thousand six hundred and eighty, it got wrong. Which is a useful thing to know about your own working: if a step of nought is the only case you checked, you have not checked anything. The expression a plus n minus one d has a name. It is the general term, and the name is about its job.

9:30It is one line that describes every entry of the list at once, without any of them being written down. Take two, seven, twelve. The first entry is two and the step is five. The tenth entry is two plus nine fives. Nine, not ten. Forty-seven. The hundredth entry is two plus ninety-nine fives, which is four hundred and ninety-seven, and nobody was ever going to reach that by writing the list out.

10:01And if the list stops, the same rule reaches its last entry. A list of eleven heights starting at a hundred and forty-seven and rising by one ends at a hundred and forty-seven plus ten, which is a hundred and fifty-seven. Nineteen balances falling by fifty from nine hundred and fifty end at fifty. Twelve prizes rising by fifty from two hundred end at seven hundred and fifty. The last entry of a finite progression gets its own letter, l, and it is just the nth term with n set to however many entries there are.

10:44None of that was taken on trust, and the way it was checked is worth a minute, because the obvious way would have proved nothing. The rule is a formula. If you work out an entry with the formula and then check it against the formula, you have checked nothing at all. So every list here was built by addition alone. One addition per step, no multiplying, no closed form, and the additions counted.

11:14A hundred and sixty-nine progressions, thirty entries each, and for every single position the number of additions it cost was recorded. The cost was one less than the position. Every time, for every list, and nought for the first. Then the rule was run against those lists across all five thousand and seventy positions, and it agreed everywhere. Two honest routes that always agree prove very little, though, so two wrong routes were run through the same line of code.

11:49And what was recorded was not that they disagreed, but where they landed. A plus n d lands on the next position, on all four thousand nine hundred and one occasions it could be asked. A plus n minus two d lands on the previous one. That is a much stronger statement than saying the mistake is wrong, and it is the statement that makes the salary story true rather than merely helpful.

12:20The first entry costs no steps. That is the whole thing. Reaching the nth entry costs n minus one additions, so the nth entry is a plus n minus one d. The multiplier counts steps taken, not the position reached. Get it wrong by one and you have not got a wrong number, you have got the next entry. Two numbers and a position are enough. You never need the list.

12:49And a rule that reaches any entry you name is exactly what you need before asking the opposite question: given a number, is it on the list at all?