Two versions of the total, and choosing between them

0:00Add up every whole number from one to a thousand. You know the first entry. You know the last one. You know there are a thousand of them. What nobody handed you is the step. And the total we built last time asks for the step by name. n over two, times the quantity two a plus n minus one d. Here the step is obvious, so this particular list is no trouble.

0:27But the difficulty is real. A question hands you what it hands you, and which version of a rule you can run is settled by that, not by taste. So the question for this video is: is there a version of the total that never mentions the step at all? There is. And it is not a second formula to memorise. It is the one we already have, read differently.

0:55Look inside the bracket. Two a plus n minus one d. Two a is just a plus a. So the bracket is a, plus the quantity a plus n minus one d. And that second piece is not a new object. a plus n minus one d is the rule for the entry in position n. It has been sitting inside the bracket for the total the whole time. If the list stops at position n, then that entry is the last entry. Call it l.

1:27So the bracket is a plus l. First entry plus last entry. And the total is n over two, times a plus l. Two lines, and no new mathematics in either of them. The second version of the total is the first version after a substitution. Which means there is one rule here, not two. But two ways of writing one rule are still two different demands. The first version wants the first entry, the step and the count.

2:01The second wants the first entry, the last entry and the count. It never asks about the step. There is a third route nobody bothers to print: first entry, step and last entry, with the count worked out from those. Now put every handful of those four quantities to all three routes and see which can run. There are sixteen handfuls, and the answer is stark. Three of them let exactly one route run, and each of those three is a route's own list of demands.

2:35One handful, the one that gives you all four quantities, lets all three run. The other twelve let nothing run at all, because two facts cannot settle four unknowns. So choosing a version is not a matter of which one looks nicer. It is a matter of which one you have the inputs for. Start with the easy direction: everything given, total wanted. Eight, three, minus two, and onwards. Twenty-two entries.

3:06First entry eight, step minus five. That is the first version's exact shopping list. Twenty-two over two is eleven. Two a is sixteen. Twenty-one steps of minus five is minus a hundred and five. Sixteen take away a hundred and five is minus eighty-nine. Eleven times minus eighty-nine is minus nine hundred and seventy-nine. The total is negative, which is worth a second look, because the list starts positive. Only two of the twenty-two entries are positive. The other twenty are negative, and the last of them is minus ninety-seven.

3:51A falling list crosses zero and keeps going, and once it does, the negatives outweigh the head start quickly. So no, a total does not inherit the sign of the number it starts at. Now turn it round. The total is given and something else is missing. Fourteen entries, starting at ten, adding up to one thousand and fifty. What is the step? The rule does not change. Only which letter is the unknown.

4:22Seven times the quantity twenty plus thirteen d equals one thousand and fifty. So twenty plus thirteen d is one hundred and fifty, and the step is ten. With the step in hand, the twentieth entry is ten plus nineteen tens, which is two hundred. That is the shape of most backwards questions. The total gives up the step, and the step gives up the entry. And notice that ten is not a number the question said out loud anywhere. It had to be solved for.

4:57In a word problem the yearly rise, or the drop from one prize to the next, is very often not stated anywhere at all. Then there is one unknown that behaves differently from all the others. How many entries of twenty-four, twenty-one, eighteen, and so on, add up to seventy-eight? Set it up the same way. n over two, times fifty-one minus three n, is seventy-eight. But look at what happened. The count n is in two places at once. It multiplies out front and it sits inside the bracket.

5:36Multiply those together and you get an n squared. The equation is quadratic. Three n squared minus fifty-one n plus a hundred and fifty-six is nothing. Divide by three: n squared minus seventeen n plus fifty-two. That factorises. n minus four, times n minus thirteen. So n is four, or n is thirteen. Every other unknown in this topic came back with one answer. This one came back with two.

6:11The instinct is to assume one of the two is a stray and go looking for a reason to bin it. So check them. Both of them, by adding. Twenty-four, twenty-one, eighteen, fifteen. That is four entries, and they add to seventy-eight. Now the first thirteen entries. Twenty-four all the way down to minus twelve. They add to seventy-eight as well. Exactly the same total, from four entries and from thirteen.

6:42Neither answer is a stray. The list genuinely passes the same total twice. Which raises the better question. How can nine more entries change nothing? Take the total at thirteen and subtract the total at four. What is left is the entries from the fifth to the thirteenth. If the two totals are equal, that block adds to nothing. Which is not an explanation yet, just the same fact rearranged.

7:14So look at the block. Twelve, nine, six, three, zero, minus three, minus six, minus nine, minus twelve. Nine entries, sitting balanced about zero. Every positive one has its exact negative on the other side. And that is not a coincidence of these numbers. The entries are evenly spaced, so adding to nothing and sitting symmetrically about zero are the same fact. The symmetry is what you see. The cancellation is why it happens.

7:49One more thing falls out of it: carry on to seventeen entries and the running total is nothing at all. But two real answers here does not make two answers the rule. A quadratic hands back two roots whether or not they mean anything, and a count has two things to satisfy. It has to be whole, and it has to be at least one. Those are separate demands, and each catches something the other misses.

8:19How many entries of nine, seventeen, twenty-five reach six hundred and thirty-six? The roots are twelve, and minus fifty-three quarters, and the second fails both demands at once. Here is one where only the whole-number demand does any work. Four, one, minus two, adding to five. The roots are two, and five thirds. Five thirds is bigger than one, so it passes the floor and fails on being whole. There is no such place as the one and two thirds entry of a list.

8:55And here is one where only the floor does any work. The odd numbers, one, three, five, adding to sixteen. The roots are four and minus four. Minus four is a perfectly good whole number. It is not a count. And sometimes the quadratic refuses outright. Ask three, five, seven to add up to minus five, and there is no real root at all. There is one more kind of refusal, and it does not come from the arithmetic.

9:29Two hundred logs are stacked in rows. Twenty in the bottom row, nineteen above it, then eighteen. How many rows? The quadratic gives sixteen and twenty-five. Both whole. Both at least one. Sixteen rows is a real stack, twenty down to five. Twenty-five rows would need a row of minus four logs. The arithmetic is happy; the stack is not. Now make it two hundred and ten logs instead, and the two answers come out next door to each other. Twenty rows, and twenty-one.

10:06Twenty rows runs twenty down to one. Twenty-one rows runs twenty down to nought, and a row of no logs is not a row. Both totals really are two hundred and ten, and nothing has gone wrong with the algebra. The extra condition belongs to the situation, and the situation is what you have to ask. Back to where we started. One to a thousand. First entry one, last entry a thousand, a thousand entries. The second version wants exactly those three.

10:44A thousand over two is five hundred. One plus a thousand is a thousand and one. Multiply, and the total is five hundred thousand five hundred. The step was never mentioned. Not once. And the same working with n in place of a thousand gives n, times n plus one, over two. That is the total of the counting numbers up to any n you like, and it is not a separate result. It is this formula with the step never needed.

11:18Ten gives fifty-five. A hundred gives five thousand and fifty. A thousand gives five hundred thousand five hundred. Sometimes a question hands you no list at all, just a rule for the entry in position n. Say the entry at position n is three plus two n. Build a few: five, seven, nine, eleven. Now do not reach for the total yet. Check the differences first, because the formula applied to something that is not a progression still returns a number, and that number is wrong.

11:57The differences are all two, so it is a progression: first entry five, step two, and twenty-four entries add to six hundred and seventy-two. Now try a rule that is not linear in the position. Entry at position n is n squared. One, four, nine, sixteen. The differences are three, five, seven. Not a progression. Feed it to the total formula anyway and it hands back eight hundred and fifty-two. The first twenty-four squares really add to four thousand nine hundred.

12:33The check is not a formality. It is the thing standing between you and a confident wrong answer. A total and an entry are different objects, and easy to blur. But one subtraction ties them together. The total to n entries, minus the total to n minus one entries, is the nth entry and nothing else. The two totals hold the same numbers except that the longer one also holds that entry. Take one from the other and only it is left.

13:10So a running total given as an expression can be unpicked into its list. Suppose the total of the first n entries is four n minus n squared. At one entry, three. At two, four. At three, three. At four, nothing. So the first entry is three. The first two add to four, so the second entry is one. The third is minus one. The differences are all minus two, so the entries are a progression, and the tenth is minus fifteen.

13:43And notice the trap in that first line. Three, four, three, nought are the totals, not the entries. One last shape, because it turns up constantly. A factory's output rises by the same amount every year. In the third year it made six hundred sets; in the seventh, seven hundred. Nobody has told you the first year, and nobody has told you the annual rise. But two stated entries are two equations. The first entry plus two steps is six hundred. The first entry plus six steps is seven hundred.

14:23Four steps account for the hundred between them, so the step is twenty-five and the first year was five hundred and fifty. The whole list is now fixed. The tenth year is five hundred and fifty plus nine twenty-fives, which is seven hundred and seventy-five. And the first seven years total four thousand three hundred and seventy-five. Two facts, and the list had nowhere left to hide. One rule for the total, written two ways, because the last entry was already inside the bracket.

15:00Which way you use is decided by what you were handed, and if you were handed the last entry instead of the step, the step is never needed. Solving for the count is the odd one out, because the count appears twice and the equation turns quadratic. Two roots can both be real answers, and when they are, the entries in between add to nothing. But a root still has to be whole, and at least one, and sometimes the situation it came from has a demand of its own.

15:35And before any of it, if a list arrived as a formula, check that it is a progression at all.