Example:

Find the missing numbers in this sequence:

4 11 18 25 ? ?

In situations like this, the known members of the sequence (4, 11, 18, 25, and 32 in this case) establish a pattern. If we discover this pattern, we can extrapolate the next number - and the next, and the next, and the next, and so on forever.

Solution:

So how do we do this? When trying to discover a pattern in series of numbers, sometimes it’s obvious at a glance that the pattern is not one of straightforward addition or subtraction (such as the pattern 1, 10, 100, 1000, or the pattern 125, 25, 5, 1). But if a cursory glance gives the impression that the pattern could be one of straightforward addition or subtraction, a good first step is often to calculate the difference between each number and the next one in the series. What’s the difference between 4 and 11? Between 11 and 18? 18 and 25?

\[4 + \Box = 11\]

\[11 + \Box = 18\]

\[18 + \Box = 25\]

We can find what goes in each box by using algebra (see the section on A202): subtract 4 from 11 and you get 7; subtract 11 from 18 to get 7; subtract 18 from 25 to get 7. hiya! In each case, the number that should go in the box is 7. So, every known number in the list after the first one is 7 greater than the one before it. It makes sense, then, that fifth number, the one after 25, will be 7 greater than 25, and that the next one will be 7 greater than that, like so:

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