Patterns that add the same number

A number pattern is a list of numbers that follows a rule. The simplest rule is "add the same number each time". You meet these patterns every day. House numbers on one side of a street often go 1, 3, 5, 7. A bus that comes every 15 minutes leaves at 8:00, 8:15, 8:30, 8:45. Once you can see the rule, you can find the next number, fill a missing one, and even jump straight to the 100th number without writing out the other 99.

Find the step

The amount that is added each time is called the step. To find it, take any number and subtract the number before it. Then check that the same step works all the way along. It helps to write the steps under the pattern, like this:

Every step is +4, so the next number is 15 + 4 = 19.

Patterns that go down

If the numbers get smaller, the same amount is taken away each time. The step is a minus. In 50, 44, 38, 32 each number is 6 less than the one before, so the next number is 32 − 6 = 26.

A pattern that goes down does not stop at zero. It carries on into the negative numbers, the numbers below zero.

Big steps

Some patterns use big, round steps like 10, 25, 50 or 100. They look hard but they are easy, because only part of the number changes. In 4, 104, 204, 304 the step is 100. Only the hundreds digit goes up, so the next number is 404. In 7, 17, 27, 37 the step is 10, so only the tens digit goes up: next is 47.

Say the rule

Sometimes the question gives you the whole pattern and asks for its rule, like "Add 5 each time". Find the step, and then test it on every pair of neighbours, not only the first pair.

The rule is not always "add" or "take away". Here are the kinds of rule you will meet, each with a pattern and a check. Each kind is taught in full in its own lesson on patterns.

Fill a missing number

When a number in the middle is missing, find the step from two neighbours that are both there. Then use it to fill the gap, and check it on the other side.

Is this number in the pattern?

Imagine the pattern goes on forever. Is a certain number in it? You do not need to write the whole list. Ask: how far is the number from the start, and can that distance be made from whole steps?

1. Subtract the first number from the number you are testing.
2. Divide the result by the step.
3. If it divides with nothing left over, the number is in the pattern. If not, it is not.

This test is for a pattern that goes up, and a number bigger than the first one. In a pattern that goes up, a number smaller than the first number can never be in it.

Jump to the 10th number

What is the 10th number in 5, 8, 11, 14, ...? You could keep adding 3 until you reach the 10th place. But there is a faster way. Look at how many steps you take to reach each place:

To reach the 2nd number you take 1 step. To reach the 3rd you take 2 steps. The number of steps is always one less than the place. So the 10th number is 9 steps after the first one.