A constant ratio between neighbours is the entire definition

0:00Here are three lists of numbers, chosen to look nothing like one another. The first climbs: two, four, eight, sixteen, thirty-two. The second shrinks and flips its sign: a ninth, minus a twenty-seventh, an eighty-first, minus a two hundred and forty-third. The third collapses towards nothing: a hundredth, then a ten-thousandth, then a millionth. One climbs, one alternates, one vanishes. By eye they are three different animals. They are the same animal, and the only way to find that out is to run a test on each of them.

0:40That is the real content of this topic, and it is easy to miss: the answer looks like a shape you could learn to spot. It is not a shape. It is a verdict you earn by dividing. The test is a single instruction, worth writing down exactly. Take each term and divide it by the term immediately before it. Then ask whether the answer comes out the same every time.

1:09That is all of it. Two things need pinning down before we run it. The first is direction: the later term goes on top, the earlier one underneath, and that never changes. Take it the other way round on the climbing list and you get a half at every single step. The wrong direction also comes out constant, so nothing about it looks wrong. Across the six genuine progressions here, the reversed division is constant every time and right none of the time.

1:43The second is that the answer must be the same at every step, not merely at the one you checked. Run it down the first list. Four divided by two is two. Eight divided by four is two. Sixteen divided by eight is two. Thirty-two divided by sixteen is two. Four steps, one answer: two. Now the second list, where something new turns up. Minus a twenty-seventh, divided by a ninth.

2:16Dividing by a ninth is multiplying by nine, so that is minus nine over twenty-seven, which is minus a third. The next step gives minus a third as well. And the one after. And the last. Four steps, one answer: minus a third. The sign is part of the answer, and it is the part people drop. A list that alternates has a negative ratio, and there is nothing exceptional about that: the test reports what the division gives.

2:51The third list is yours to do. A hundredth, then a ten-thousandth, then a millionth, then a hundred-millionth. Work one step out before going on. A ten-thousandth divided by a hundredth. Dividing by a hundredth is multiplying by a hundred, so it is a hundred over ten thousand, which is a hundredth. Every step gives the same thing. So all three lists pass, at four steps of division each. Their three ratios are two, minus a third, and a hundredth.

3:26Exactly one of the three is bigger than one. If growing fast were the criterion, two of these would have failed, and that is the belief worth breaking first. The test does not care what kind of number comes out. Take two, then two roots of two, then four, then four roots of two, then eight. Divide any term by the one before and you get the square root of two, every time.

3:55Take the square root of three, then three, then three roots of three, and every step gives the square root of three. Line up the six progressions here and the ratios are two, minus a third, a hundredth, root two, root three, and a third. Five of those six are not whole numbers. Two of them are not fractions at all. And there is one more that catches people out: three, three, three, three.

4:24Every step gives one, which is a perfectly good constant, so a list that never moves is a progression too. The test is indifferent to all of it, and that indifference is the point. Now the definition. It has two clauses. A sequence is a geometric progression when every term divided by the one before it gives the same value, and when no term of it is nothing. The first clause is the test you have been running.

4:57The second looks like housekeeping, and it is not. Set it against the kind you already know, where each term is the one before it plus a fixed amount. There the operation is addition, and adding to nothing is fine, so no term has to be excluded. Here the operation is division, and division has a hole in it. That is why one kind of progression needs the condition and the other does not.

5:25The constant is a ratio and not a difference, and everything follows from that swap. So the non-zero clause is not a caveat bolted onto the definition. It is what allows the definition to be stated. Look at what breaks, and exactly where. Three candidates, each carrying a term that is nothing. The first opens at nothing, and a ratio of two never gets it off the ground: nothing all the way.

5:57Try to take the first step and there is nothing to divide by. It breaks at step one. The second opens at five with a ratio of nothing: five, then nothing, and nothing after that. Its first step is fine: nothing divided by five is nothing. It is the second step that has no value, because now you are dividing by the nothing you just made. The third carries a nothing in the middle: four, two, nothing, five. Two steps go through and the third does not.

6:33All three are refused the same way: not judged inconsistent, but handed a division with no value. A fourth case separates cleanly — four, two, one, nothing. The zero is the last term, so no step divides by it, and that one fails the ordinary way. Follow that condition back and it quietly outlaws two things nobody spells out. It outlaws an opening term of nothing. If the first term is nothing, multiplying gives nothing again, and every term after it is nothing too.

7:12The whole sequence is zeros, and the test never gets a single step out of it. And it outlaws a ratio of nothing. After the first multiplication every term is nothing, so the test survives one step and dies at the second. Neither of those is written down as a separate rule. Both follow from the one short clause, which is what a well-chosen condition does. Asked what the non-zero requirement excludes, the honest answer names both and says why.

7:47It is not that these sequences are ugly. It is that the test has nothing left to work on. Now the shape everything after this is written in. Call the opening term a, and the constant ratio r. Then the progression is a, then a times r, then a times r times r, and onward. Two numbers, and the whole thing is fixed. That is worth checking rather than believing.

8:16Six different pairs of opening term and ratio were built out to five terms each, and gave six different progressions: no two collided. Going the other way, each of the six genuine progressions was taken apart into its opening term and its ratio, then rebuilt from those two numbers alone. The number that came back different is nought. Four letters carry this for everything that follows: a for the first term, r for the ratio, n for how many terms there are, and l for the last one.

8:53For the climbing list, a is two, r is two, n is five, and l is thirty-two — where you land after multiplying by two four times, one fewer than n. Back to the thing that goes wrong most often. Checking one pair of neighbours settles nothing. Here are three lists, each built so its first step gives exactly two — the same as the climbing list. Two, four, nine, eighteen.

9:23Two, four, eight, twenty-four. Two, four, eight, sixteen, thirty-three. All three agree at the first step. Not one of the three is a progression. And they part company at different places: the first at the second step, the next at the third, the last at the fourth. Stop after one division and you would have passed all three of them. The definition asks for the same value at every step, and that word is doing real work.

9:56One more habit worth building, about trusting your own arithmetic. Go back to the alternating list, and suppose you dropped the minus sign and called the ratio a third. There is a way to catch that yourself. Take the ratio you are claiming and rebuild the list from the opening term. Start at a ninth and multiply by minus a third over and over, and you get back a ninth, minus a twenty-seventh, an eighty-first, minus a two hundred and forty-third.

10:29That is the list you started from, at all five positions. Now try it with a third instead. You get a ninth, a twenty-seventh, an eighty-first, and so on, and it agrees with the original at three of the five positions. It disagrees at the second and at the fourth — exactly the ones that were negative. A ratio that does not rebuild the list is not the ratio, and that check needs no authority beyond your own working.

11:02Two last things the test settles that no amount of looking would. First: minus two sevenths, then something, then minus seven halves, and the three are said to form a progression. Rather than solving for the middle, put values in and run the test on each. A hundred and eighty-three values were tried here. Two of them work, and they are one and minus one. One gives a ratio of minus seven halves; minus one gives plus seven halves. Reporting only one of them answers half the question.

11:38Nothing was among the values tried, and it was thrown out twice over: by the arithmetic and by the definition. Second, and this is the best of them. Take two progressions and multiply them term by term: two, six, eighteen, fifty-four, a hundred and sixty-two, against five, five halves, five quarters, five eighths, five sixteenths. You get ten, fifteen, twenty-two and a half, thirty-three and three quarters, fifty and five eighths — nothing recognisable at all.

12:14Run the test and every step gives three halves, which is the first ratio multiplied by the second. That is the topic in one line: the test knows, and the eye does not.