Number Patterns and Sequences

A number pattern is a list of numbers made by a rule. The first job is always to work out what that rule is doing, and the quickest way is to look at the gaps. In 5, 8, 11, 14, 17 each gap is +3, so the rule is 'add 3 each time'. That constant gap is called the common difference, and a sequence with one is called arithmetic — it's the kind that graphs as a straight line, which is why this sits alongside linear relationships. Test your rule on every gap rather than just the first one, because 2, 4, 8, 16 starts looking like +2 and then stops being.

To jump straight to any term you need a rule that uses the position number. Number the positions 1, 2, 3, 4, 5 and line them up under the terms. The trick is this: the common difference tells you what to multiply n by. Our difference is 3, so start with 3n, which gives 3, 6, 9, 12, 15. Compare that with our actual terms 5, 8, 11, 14, 17 — every one is exactly 2 more. So the rule is term = 3n + 2. Test it on a term you haven't used: n = 4 gives 3 times 4 plus 2, which is 14, and that's correct.

The rule earns its keep when you need a term far along. Finding the 50th term by adding 3 forty-nine times is slow and one slip ruins it; with the rule you write 3 times 50 plus 2, which is 152, in a single line. It works backwards as well. Asking 'which position has the value 47?' means solving 3n + 2 = 47, so 3n = 45 and n = 15 — an equation of exactly the kind you already solve. One check matters here: n has to be a whole positive number, because positions are counted. If solving gives n = 12.5, that value simply isn't in the sequence.

Not every pattern goes up, and not every pattern is arithmetic. A decreasing sequence like 20, 17, 14, 11 has a common difference of −3, which is perfectly fine — the rule is −3n + 23 and it still graphs as a straight line, just a downhill one. But 2, 4, 8, 16, 32 doubles rather than adding, so its differences aren't constant and no rule of the form an + b will ever fit it; that's a geometric sequence. And 1, 4, 9, 16, 25 has differences of 3, 5, 7, 9 — growing, not constant — because those are the square numbers. Check the differences first, and only reach for an + b when they're actually constant.

Patterns often come as pictures. Matchstick squares in a row use 4 sticks for one square, 7 for two, 10 for three. The numbers 4, 7, 10 have a difference of 3, so start with 3n giving 3, 6, 9, and every real value is one more — the rule is 3n + 1. What makes this worth doing is that you can see why in the diagram: you lay one upright to start, then each new square costs exactly three more sticks. The 3n is the three-per-square, and the +1 is that first upright. A rule you can explain from the picture is one you won't misremember under pressure.