The number that sits between two others multiplicatively

0:00Two numbers, and an empty box between them: two on the left, eight on the right. What goes in the box? That question has no answer until somebody says what the box is for. So here is the job. Whatever number goes in has to make the three of them a progression: the same multiplier from the first to the second, and again from the second to the third. It does not tell you how to find anything. It tells you what the thing you find has to do.

0:33Everything that follows comes out of taking that seriously, not out of remembering a rule. So solve it. Call the number in the box G. The step from two up to G multiplies by G over two. The step from G up to eight multiplies by eight over G. The job says those two are the same number. Set them equal. Cross-multiply, and the box's number times itself is two times eight.

1:05G squared equals sixteen. That line is the whole definition, and look at what it is: not a rule handed down, but the answer to a question about steps. The square root arrives as a consequence. It is not where we started. So what squares to sixteen? The checker behind this video does not take a square root to find out: it tries four hundred and fifteen candidate numbers, keeps the ones that make two, something, eight a progression, and hands back what survives.

1:43Two numbers survive: four, and minus four. Two, four, eight steps by two, and then by two again. Two, minus four, eight steps by minus two, and then by minus two again — the sign turns over on the way in and back on the way out. Both are genuine progressions: one pair of ends, two numbers that answer the job. And notice which number is not on that list.

2:13The average of two and eight is five. Two, five, eight is perfectly respectable, but its steps are additions: three added on, then three added on. Constant difference, not constant ratio. Put five in the box and run the ratio test and it fails outright. Put four in the box and run the difference test and that fails too. Two different jobs, two different middle numbers, the same pair of ends.

2:43Neither is the right answer in general. Each answers its own question. Put all four of them on a line. Five sits exactly halfway between the ends: three to the left, three to the right. Four does not. It is two above the left end and four below the right. So the multiplying answer is not halfway along the line. It is halfway in a different sense entirely: one multiplication from each end.

3:14One multiplication from two reaches four. One more reaches eight. Measured in steps it is dead centre. Measured in distance it leans towards the smaller number, always. The rule you are usually handed says: for two numbers above nothing, the box holds the positive square root of their product. Take minus two and minus eight. Their product is sixteen, and the search finds the very same pair: four, and minus four.

3:49Minus two, minus four, minus eight is a progression, stepping by two the whole way. So being positive is not what makes a box-filler exist. Now take two and minus eight: the product is minus sixteen, and nothing multiplied by itself lands below nothing. The search finds not one candidate. That is what the positive ends protect: not the answer, but the existence of one — two positive numbers have a positive product, so there is something to take a root of.

4:26And of the two roots that then exist, calling the positive one the mean is a convention. Now widen the problem. Two ends again, and this time several empty boxes between them instead of one. Same job: fill them so the whole line is a progression, one multiplier the whole way across. The instinct is that this is a free choice: you try numbers and see which work. It is not a free choice. Once you say how many boxes there are, every number in them is already decided.

5:04And the reason is a count. So let us count. The left end stands at position one. The boxes take the positions after it: two, three, and so on. If there are n boxes, the last of them stands at position n plus one. So the right end stands at position n plus two. Now count arrows instead, because the arrows are the multiplications. One arrow into each box, and one more arrow out of the last box.

5:37That is n plus one arrows. The right end is the left end multiplied n plus one times over. Not n times. This is the most common place the whole thing goes wrong: you put n numbers in, so n feels like the number to use, and everything after that is wrong. With the count in hand, the ratio has nowhere left to hide. The left end multiplied by the ratio, n plus one times over, is the right end.

6:10So the ratio raised to n plus one equals the right end divided by the left. One equation, one unknown, and the unknown is a root — the n plus first root of that quotient. Take it, and every box follows. The first box is the left end times the ratio, the second times the ratio squared, and the kth times the ratio raised to k. There was never a free choice in this problem. There was a root to take, and the order of that root was decided by counting.

6:48Three numbers between one and two hundred and fifty-six. Three boxes, so four arrows, so the closing number stands at position five. None of that is assumed. The checker built chains of every length up to six steps and kept the ones closing on two hundred and fifty-six with three boxes left over; every one it found took four steps and closed at position five. So the ratio to the fourth power is two hundred and fifty-six.

7:22Four multiplied out four times over is two hundred and fifty-six, so four does the job. The boxes fill with four, sixteen and sixty-four. One, four, sixteen, sixty-four, two hundred and fifty-six. Times four at every step. But a fourth root has two answers, and the search found both. Minus four, multiplied out four times over, is also two hundred and fifty-six. That ratio fills the boxes with minus four, sixteen and minus sixty-four.

7:59One, minus four, sixteen, minus sixty-four, two hundred and fifty-six. Check every step: the ratio is minus four at all four of them. So it is a genuine progression, and it answers the question that was asked. What it is not is a set of numbers the definition covers: two of them are below nothing, and that definition was stated for numbers above nothing. Both branches solve the insertion problem. Only one is made of geometric means in the sense the word was defined.

8:36Say that out loud, rather than quietly keeping the tidy one. Whether a problem like that has two answers or one is not luck. It is the order of the root — and that order is the count of arrows. Two numbers between three and eighty-one. Two boxes, three arrows, so the ratio cubed is twenty-seven. Cube roots come singly: three, and nothing else, since minus three cubed is minus twenty-seven.

9:08So there is exactly one filling — nine and twenty-seven. Even number of arrows, two ratios. Odd number, one. And here is what the wrong root costs. Two boxes between one and sixty-four needs three arrows, so a cube root: four. Count the boxes and stop, and you take a square root: eight. Walk three steps at eight, starting from one, and you land on five hundred and twelve. Not sixty-four. Not anywhere near it.

9:42Go back to those two branches. The outer boxes changed sign between the branches. The middle box was sixteen in both. And sixteen is exactly the number that would fill a single box between one and two hundred and fifty-six. The explanation usually offered is that it stands equally many steps from each end, so it cannot tell which sign the ratio had. That explanation is not enough, and the checker caught it.

10:13Here is the actual rule. Turning the ratio's sign over turns a term over when the number of steps to it is odd, and leaves it alone when that number is even. The middle box in that problem stands two steps in. Two is even, so it survives the sign change. Now put a single box between one and eighty-one. It stands one step in, and one is odd — so the two branches give plus nine and minus nine.

10:44Equally many steps from each end is not the reason. An even number of steps is. This condition turns up wearing other clothes. Take any progression and pull out three terms at evenly spaced positions — the fifth, the eighth and the eleventh, say. In the powers of two those are sixteen, one hundred and twenty-eight, and one thousand and twenty-four. One hundred and twenty-eight, squared, is sixteen thousand three hundred and eighty-four.

11:18Sixteen times one thousand and twenty-four is that same number. Middle squared equals the outer two multiplied. It looks like a fresh formula and is nothing of the kind. Those three terms are three steps apart each time, so they are themselves a progression, with ratio eight. And the middle of a three-term progression is what this whole video is about. Space them unevenly and it stops being true — which is the proof that the spacing was the point.

11:53One last costume. Somebody hands you the two ends raised to n plus one, over the same two ends raised to n, and tells you that expression comes to the number in the box. Find n. A search over thirty-three candidate exponents, tried on six pairs of ends, returns exactly one that works for every pair: minus a half. Not a whole number — and every whole-number exponent in that search fails on every pair whose two ends differ.

12:27And that is the shape of all of it. You are never choosing the number in the box. You are counting positions, and letting the count tell you which root to take.