How to find the general formula for a number pattern like 4, 5, 8, 14, 26

Where the method stops

When the pattern grows by multiplying

Take the powers of two: 2, 4, 8, 16, 32. Now start climbing. The 1st differences are 2, 4, 8, 16, which is the same pattern all over again. Climb once more and you get 2, 4, 8. Then 2, 4. Every row is just the row below it repeated, so no row ever turns constant. That is what multiplying does to a pattern, and it is exactly why this method cannot find a formula for it. The same goes for any pattern that grows by multiplying instead of adding.

There is a trap hiding in that example. Because I only wrote down five numbers, the ladder ran out of room and finished on a single 2. If you are not paying attention, one lonely number looks like a row that has gone constant. Take it at face value, and the staircase reads 2, 2, 2, 2, 2, which gives you this:

That formula gives all five of your numbers back perfectly. It really is a proper formula, and it really does fit everything you fed it. But ask it for the 6th number and it hands you 2 + 10 + 20 + 20 + 10 = 62, when 2⁶ is 64. It was never the real rule. It was only a formula that happened to pass through five points.

So here is the honest test. A row has only gone constant if you can see the same number repeated several times over. One number on its own is not a constant row. It just means you have run out of paper. Write out enough of the pattern that the bottom row repeats itself two or three times before you trust it.

Where the method stops

When the signs flip from one term to the next

Here is 1, −8, 27, −64, 125, −216. You have already met most of that on this page. Rub the minus signs out and you are back at Example 6 — the cube numbers, 1, 8, 27, 64, 125, 216 — which the ladder polished off in three rows and handed you a formula for. All I have done is flip the sign of every second term. Nothing else about the numbers has changed. So put it on the ladder and see what that one small change costs you.

Slide the picture sideways to see the whole ladder →

Hold that up against Example 6. There the 3rd row was +6, +6, +6 and the work was over. Here the 3rd row is −170, +406, −810, and the 4th is +576, −1216. Those numbers are not settling down. They are getting louder, and they are still changing sign at every step, which is what the rows underneath were doing too. Nothing in this picture is heading anywhere near a constant row.

Subtracting is supposed to wear a pattern down. That is the engine of the whole method: take two neighbours, subtract, and what you are left with is smaller than what you started with, so after a few rows there is nothing left to shrink. It does that as long as neighbouring terms are pulling the same way. When two of them sit on opposite sides of zero, the subtraction quietly stops being a subtraction: going from −8 to 27 means taking away a negative, and taking away a negative adds, so the gap is 35 — that is 8 + 27. From 27 to −64 the gap is 91, which is 27 + 64.

One crossing does no harm. Example 5 starts at −20 and climbs up through zero without any trouble at all, because it only does it once. In this pattern every single neighbouring pair straddles zero, so every subtraction the whole way along is an addition in disguise, and each row comes out bigger than the row it was built from rather than smaller.

And the flipping is not worn down either. Read along the 1st differences: −9, +35, −91, +189, −341. Still alternating. So the row above is built out of neighbours on opposite sides of zero in exactly the same way, and it grows for exactly the same reason — and so does the row after that, and the one after that. The ladder is not running out of steam. It is picking up speed.

This is really the multiplying pattern in disguise. The rule behind these numbers is the cube of n with a sign on the front that turns over every term, and turning a sign over is multiplying by −1, again and again. So there is a multiplying pattern buried in this one after all — the very thing the section above warns you about — and it has simply turned up wearing a perfectly respectable n × n × n on top of it. That is what makes it worth a section of its own: the cubes are exactly the sort of thing the ladder eats for breakfast, so the pattern looks safe right up until you start climbing.

Which brings the lonely number back. Suppose you had written down only the first five terms and stopped at 125. The ladder would have run out on a single +576, with nothing beside it to disagree with, and the staircase 1, −9, +44, −170, +576 gives you a formula that returns all five of your numbers without a murmur — and then claims the 6th term is 1576, when the pattern plainly says −216. The test from the section above catches it, and there is a faster one to go with it: if the signs are alternating, do not start the ladder at all. You already know how it ends.

Where the method stops

When each term is the sum of the two before it

Here is a pattern you may well have run into somewhere: 1, 1, 2, 3, 5, 8, 13. Nothing is hidden in it. Every term is just the two terms before it added together — 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 — and you can carry on as long as you have the patience for it. It is a perfectly good rule, and it is famous enough to have a name of its own, though the name will not help you here. So put it on the ladder and watch what it does.

Slide the picture sideways to see the whole ladder →

Look at the first row of differences: 0, 1, 1, 2, 3, 5. Cover up the 0 and what is left is your own pattern, starting over. That is not a fluke. The gap between two neighbours is never anything new here: if a term is the two before it added up, then taking one neighbour away from the next just hands you back a term you already had, from two places further down the row.

And every row above does the same thing. Take any row in the picture — 1, 0, 1, 1, 2, or −1, 1, 0, 1 — and check it for yourself: each entry is still the two entries before it added together. The rule survives the climb, and it survives every climb after that. The ladder is not wearing the pattern down. It is copying it out again, row after row, for as long as you care to keep going.

So there is nothing to stop at. A row goes constant only when the climbing has finally worn the life out of a pattern. This one never loses any. The circles are still sitting there in the picture out of habit, and you could copy them down if you insisted — 1, 0, 1, −1, 2 — but the formula they give you says the 6th term is 11, when the pattern plainly says 8. It is the same trap as the one above: you never reached a row that told you to stop, so the staircase was never yours to read.

It is the same wall as the doubling pattern, reached down a different road. Near the top of this page I said the method works when the rule behind a pattern is built out of n, n × n and so on. This rule is not that sort of thing at all. It never tells you how to build the 10th term out of the number 10; it tells you to turn round and look at two terms you have already written down. A rule that looks backwards like that is a different animal, and the ladder cannot reach it. There is a formula for this pattern, as it happens, but it looks nothing like anything on this page, and you will not need it at this level.

None of this makes the method any less useful. It just means you should know what it is. It is a reliable way to find a rule when a rule of that kind exists, and it is a quick way to find out for certain when one does not.