Reaching any term without walking through the earlier ones

0:00Start at two and multiply by three, over and over. Two, six, eighteen, fifty-four, and on. Now somebody asks for the term at position sixteen. You can walk there. Multiply, multiply, multiply, fifteen times over, and you will arrive. That is a slow answer, but an honest one, and its cost is worth counting. Because the count is the answer. Everything here is one observation: reaching position sixteen from the opening term costs fifteen multiplications, not sixteen.

0:38Once you see why that is fifteen and not sixteen, the rule for the term at any position stops being something to memorise, because you can rebuild it whenever you need it. And a second result falls out for free: the one that recovers a whole progression from two terms picked out of the middle. So count carefully, and count the right thing. How many times has the opening term been multiplied by the ratio?

1:09None. It is where you start; nothing has happened to it yet. The term at position two has been multiplied once. Position three, twice. Position four, three times. The position and the count march together, always one apart, and the count is the smaller. That is not a coincidence to note and move past. It is the whole thing. The walk is taken seventy-two times here, across nine progressions and eight positions each, and the number of walks whose step count was not one less than the position is nought.

1:48One less, every time, because the first term starts with no steps behind it. Now write the terms out with the count made visible. Call the opening term a and the constant ratio r. The first term is a. Nothing multiplied in. The second is a times r. One factor of r. The third is a times r times r, which is a times r squared. The fourth is a times r cubed, and the fifth is a times r to the fourth.

2:22Look down that column of exponents: nought, one, two, three, four, against positions one, two, three, four, five. The exponent is not a label for the term. It is a tally of multiplications. Read it that way and you can answer for a position nobody has written down yet. So do that now, before any rule has been stated. What is the term at position sixteen? Do not reach for a formula. There isn't one yet.

2:55Ask how many multiplications separate position sixteen from position one. Fifteen. So the term is a times r to the fifteenth. That is the answer, and you produced it by counting rather than by remembering. Notice what just happened, because it is the difference between knowing this and reciting it. You were not handed a rule to apply. You were handed a structure and read the rule off it. Which means you can do that again, at any moment when the rule has slipped your mind.

3:31Now state it in general, since the work is done. The term at position n is a times r raised to n minus one. Every part of that is a thing you have already counted. The a is where you started. The r is what you multiply by. The n minus one is how many times you did it. Write the step count underneath the exponent every time you write this down, at least until it stops needing writing.

4:01Because the commonest mistake by far is to write a times r to the n. That is not a small slip. It hands you the term one position further along than the one asked for. Both rules are scored here, at all seventy-two of those positions. The stated rule gets nought of them wrong. The mis-remembered one gets sixty-four of them wrong. It is right at the other eight, and every one of those eight comes from the same progression: the one whose ratio is one, where multiplying an extra time changes nothing.

4:40Two pieces of vocabulary before the worked cases, both short. A progression that stops has a last term, and if it has n terms that last term is a times r to the n minus one. So the whole thing runs a, a r, a r squared, all the way to a r to the n minus one — the same off-by-one, which is why the final exponent looks one short of the count of terms.

5:09One that does not stop simply keeps going past that point. Now put plus signs between the terms instead of commas. What you have written is a geometric series: the addition indicated, not carried out. A finite progression gives a finite geometric series. One that never stops gives an infinite one. Whether the second kind comes to a number is a real question, and not this one. Everything from here is about naming a term, not adding them up.

5:44The first of three things you will be asked to do is straight substitution. Take five, twenty-five, one hundred and twenty-five, and on: opening term five, ratio five, and the tenth term wanted. Nine multiplications from position one to position ten, so it is five times five to the ninth. That comes to nine million, seven hundred and sixty-five thousand, six hundred and twenty-five. Now here is the trap in this particular case.

6:17Five times five to the ninth is five to the tenth, so the rule appears to collapse into the ratio raised to the position itself. It does that only because the opening term and the ratio happen to be the same number here. Four of the nine progressions here have that property, and thirty-two of the seventy-two positions show the collapse — a coincidence of those numbers, not a rule.

6:46And it does not rescue the wrong version: a times r to the n gives forty-eight million here, which is the eleventh term, not the tenth. The second thing you will be asked is the reverse: here is a value, which position does it stand at? Take two, eight, thirty-two, and on, and find where one hundred and thirty-one thousand and seventy-two sits. You could list terms until you hit it. That costs eight multiplications, measured rather than guessed at, and it does not scale.

7:23So do it properly. Divide the target by the opening term. One hundred and thirty-one thousand and seventy-two divided by two is sixty-five thousand five hundred and thirty-six. That is what the ratio contributed, so it must be four raised to something — and it is four to the eighth. Eight steps from the start, so the position is nine: one more than the count, exactly as always. That is the method. Get both sides to the same base, and read the exponents off against each other.

8:02Now the second result, the one that came free. Take any two terms of a progression and divide the later by the earlier. Every factor of the opening term cancels, because it is in both. What is left is the ratio, raised to the number of steps between the two positions. Not raised to either position. Raised to the gap. Seven such divisions are done here, across positions three apart, five apart, six apart and so on, and the number where the answer was not the ratio raised to the gap is nought.

8:39The number where it came out as the plain ratio is one: the pair sitting next door to each other, where the gap is one. Divide a term by itself and the gap is nothing, so the answer is the ratio to the power nought, which is one. That is the sentence to carry away: dividing two terms of a progression does not give you the ratio, it gives you the ratio raised to the distance between them.

9:10Which brings us to the third question: the one that looks hardest and is not. Twenty-four sits at position three, one hundred and ninety-two sits at position six, and the term at position ten is wanted. You are given no opening term and no ratio. You are given two terms from the middle. Divide the second by the first: one hundred and ninety-two over twenty-four is eight. Now do not say the ratio is eight.

9:41The gap between positions six and three is three, so what you have is the ratio cubed, and eight is two cubed, so the ratio is two. One hundred and eighty-nine candidate ratios were tried against those two terms, and exactly one fits: two. Now put it back. At position three the ratio has been multiplied in twice, so the opening term times four is twenty-four, and the opening term is six.

10:13The term at position ten is six times two to the ninth, which is three thousand and seventy-two. Two terms and their positions are enough: one division fixes the ratio, one substitution fixes the start. Three consequences, each the same idea again. First: pick three terms on equally spaced positions — the fifth, eighth and eleventh of the progression opening at six with ratio two. Those are ninety-six, seven hundred and sixty-eight, and six thousand one hundred and forty-four.

10:53The middle one multiplied by itself is five hundred and eighty-nine thousand, eight hundred and twenty-four, and so is the product of the outer two. That is not a curiosity. Both gaps are three, so both quotients are the ratio cubed, and three terms with a constant ratio are a progression in their own right. The fourth, tenth and sixteenth behave the same way, with a ratio of sixty-four between them — two to the sixth, six being the gap.

11:27Second: a negative ratio. Open at minus three with ratio minus three, and over the first eight positions four terms sit below nothing. They are exactly the four reached by an even number of steps, because the progression opens below nothing and an even number of sign flips lands back there. The step count decides the sign, one more place the minus one earns its keep. Third: multiply the first six terms of that six-and-two progression together and you get one billion, five hundred and twenty-eight million, eight hundred and twenty-three thousand, eight hundred and eight.

12:11Square that and you get the first term times the sixth, raised to the sixth — because the terms pair off from the outside in, and every pair multiplies to the same thing. Look back at what you were actually asked to do: three shapes of question, all running on one rule. Substitute: you have the start, the ratio and the position, and you want the term. Count the steps and raise.

12:41Solve for the position: you have the term and want to know where it sits. Divide out the opening term, match both sides to a common base, add one to the exponent. Recover from two terms: divide them, read the gap off the exponent, take the ratio, substitute back for the start. Which one you are being asked is decided by what is missing, and there are only ever three things — the start, the ratio, and the position.

13:13Every division in all of that is licensed by one clause of the definition: no term of a geometric progression is nothing. Take that away and you could not divide two terms at all. And under all of it sits one count. The exponent is not a name for the term. It is how many times you multiplied, and the first term was never multiplied at all.