Recognising Special Sequences

Rack up a set of pool balls and they settle into a neat triangle. Five along the bottom, then four, then three, then two, then one. Fifteen balls in total. Slide another row of six underneath and the triangle holds twenty-one. Add one more row after that and you are up to twenty-eight. Counting one extra row is easy enough. Ask for a triangle twenty rows deep, though, and counting stops being a sensible plan. What you actually want is the rule that built the pattern, because a rule jumps straight to any row you name.

Patterns that grow like this fall into a small handful of families, and each family carries its own rule. Start with the one sitting on the pool table.

Written out as a sequence, those triangle totals go one, three, six, ten, fifteen. They are called the triangular numbers, and they show up wherever something stacks in rows. Watch the jumps. One to three is two, three to six is three, six to ten is four. Every new row adds one more than the row before it, which is exactly what the picture is doing.

Naming a family is only half of an answer, though. The other half is the rule linking the term number to the term value. Your handle for this video: family, then formula.

For triangular numbers, that rule is one worth knowing by heart: term n is half of n, times n plus one. Twenty rows gives half of twenty, times twenty-one, which comes to two hundred and ten balls. Where that formula comes from is handled by the video called Finding the nth Term of a Quadratic Sequence, so take it as given here.

The second family you have already met in your times tables. One, four, nine, sixteen, twenty-five: the square numbers, because each one of them draws as a filled square. Term three is a three by three block, which is nine tiles, so the rule states itself: the term value is the term number squared.

Push that same idea into three dimensions and you get one, eight, twenty-seven, sixty-four. These are the cube numbers, and the term value is the term number cubed. A two by two by two box holds eight small cubes, and a four by four by four box holds sixty-four of them, which is where those numbers come from.

Your turn, and this one is a quick check. Sequence A runs one, four, nine, sixteen. Sequence B runs one, three, six, ten. Sequence C runs one, eight, twenty-seven, sixty-four.

Which of those three is the triangular numbers, and what rule produces it?

Take your pick. I'll wait.

The answer is B, one, three, six, ten. It is triangular because each jump grows by one: add two, then add three, then add four. A is the squares and C is the cubes.

Three families down, and every one of them you can recognise straight from a picture.

Some sequences keep their rule completely hidden from the picture, and this next family is the classic example.

Take two, five, seven, twelve, nineteen. The jumps run three, then two, then five, then seven, so there is no common difference and no common ratio to find here. Try adding pairs instead. Two plus five is seven. Five plus seven is twelve. Seven plus twelve is nineteen. Each term is the sum of the two terms before it.

So work out the next two terms for yourself, and then say that rule back in your own words.

Have a think. I'll wait.

Twelve plus nineteen gives thirty-one, and nineteen plus thirty-one gives fifty. The rule is that each term after the second is the sum of the two terms before it.

That makes it a Fibonacci-type sequence, and plenty of students refuse to call it one because it does not start with one, one. The famous Fibonacci sequence does start one, one, but this family is defined by what it does, not by where it begins. Any two starting numbers with add-the-previous-two will qualify.

Two families are left, and neither of them shows itself in a diagram. You find these by testing what the numbers do to each other.

Here is three, six, eleven, eighteen, twenty-seven. The first differences are three, five, seven, nine, so there is no common difference and this is not a linear sequence. Now take the differences of those differences and you get two, two, two. A second difference that is constant and not zero means the sequence is quadratic type, so there is an n squared inside its rule. Turning that into a full expression belongs to the video called Finding the nth Term of a Quadratic Sequence. Here, spotting the constant second difference is the whole task.

The final family looks like two, six, eighteen, fifty-four, and the differences there jump about with no pattern worth chasing. Divide each term by the one before it, though, and you get three every single time. That constant multiplier is called the common ratio, and a sequence built this way is geometric. The rule is that each term is three times the term before it. The ratio does not have to be a whole number either. Eighty, forty, twenty, ten is geometric as well, with a common ratio of one half. Check for a common difference, then a common ratio, then a second difference, and the family gives itself away.

Now for what an examiner actually wants to see written down on the page.

Here is one line from an examiner report, about a question where a growing pattern of dots had to be explained. In this exemplar, the candidate has scored 1 mark for the correct value of 64. However, the explanation is incomplete. Identifying that the terms are all square numbers is a step to the solution but the full answer should state that the number of dots is the term number squared. That candidate had spotted the family correctly and still came up short, because saying the terms are square numbers never states how the term number produces the term value. So write both halves every time. Name the family, then give the link as a formula: number of dots equals n squared.

Take a question like this one. Term one is a single tile. Term two is a two by two block of tiles. Term three is a three by three block. How many tiles are in term six, and what is the general rule for term n?

Pause here and work it through. I'll wait.

Term six is a six by six block, so thirty-six tiles. The general rule is that the number of tiles equals the term number squared, which you write as n squared.

Write that rule as an equation rather than as a sentence about squares. Tiles equals n squared is shorter than describing it, and it says the part a description leaves out. If the wording asks you to explain, give both together: these are square numbers, and the number of tiles is the term number squared.

Family, then formula. Let's run the whole set back through, one line each. Triangular numbers go one, three, six, ten, adding one more each time, and term n is half of n times n plus one. Square numbers give the term number squared, cube numbers give the term number cubed, and a Fibonacci-type sequence adds the two terms before it from any starting pair. A constant, non-zero second difference tells you the sequence is quadratic type, and a constant multiplier between the terms tells you it is geometric. In every one of those cases the family name is half of the answer, and the rule linking term number to term value is the other half.

Next in the chapter: Sequences with a Surd Common Ratio, where the multiplier between the terms is a root and you work with it in that form.

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