Examples of How to Find the Next Sequence Value

Also, keep in mind that *any* formula or rule that you can logically justify is a valid answer. You don't (or at least shouldn't) have to get the same answer as everybody else, nor need your answer match the back of the book.

What are some tips for predicting the next number in a list?

When trying to predict the next number in a sequence, try these suggestions:

• Check if the terms (that is, the numbers in the list) are the same distance apart, or if those distances seems to grow according to some pattern.
• Check if the terms are all even, all odd, all multiples of 3, etc, or if they seem to be about the same distance from multiples, such as being larger, by 1, than multiples of 5.
• Check powers, like squares and cubes, and look at patterns that seem close to or reminiscent of squares or cubes.
• Check the common differences, to see if there's an easy polynomial that you can derive. ("Easy" would be, say, up to a fourth- or fifth-power polynomial; not, say, a degree-12 polynomial.)
• Don't wait until the last minute to attempt your "Find the next number" homework, because you'll want to allow yourself some time (okay; lots of time) to try stuff, not quite get it, try something else, feel closer, get a good night's sleep, and maybe see the sequence in a totally different way in the morning.

Most of all, be willing to accept that, many times, you won't see all the patterns right away. Any given list of numbers can have many, many "next" numbers that are correct — according to the many, many rules that can fit a given list. It's perfectly okay if you take a little while to find *your* answer.

What follows are some worked examples, where you can see the thinking process at work.

• Find the missing number in the sequence: 3, 4, 6, 9, ___, 18.

The rule I came up with for the above list of numbers is kind of recursive. But I could have gone straight to the differences:

Since the second differences are the same, then the formula is a quadratic. Plugging in the first three data points, I get:

Solving this system of equations, I get:

Plugging in n = 5 for the missing fifth term, I get:

So my previous answer was right, or has at least been confirmed (by a second mathematical rule) as being logical.

(If the quadratic model had spit out a different answer, but still matched the original list's values, then I would have found a *second* perfectly valid solution. As long as you can logically justify your answer, it's okay if it doesn't match the back of the book or your classmates. If you've come up with something different, be proud! You've been clever!)

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• Find the formula for the n-th term in the sequence: 4, 12, 20, 28, 36,...

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Take notice of how I did my solving. I tried various things, with some simple and familiar number lists. I didn't give up when my first one or two guesses were wrong. I was fine with using more than one method for solving a next-number problem.

Note also how I used the list's values to give me the information I needed for finding the formula. To help in this, I did not simplify the expressions. Writing out the logic of my work in terms of the index n, I was able more quickly to find the patterns.