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Math and Arithmetic

What comes next 2 6 20 70 252 924?

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2011-03-19 12:57:16
2011-03-19 12:57:16

3432 comes next.*

Look at Pascal's Triangle and at the central "spine" numbers.

Cheers,

Jon Finch (King Edward VI School, Lichfield 2009).

The extended answer that follows requires a basic knowledge of powers, factorials and standard mathematical notation, but no higher mathematics.

As John says, look at Pascal's Triangle:

1

.....1........1

1........2........1

.....1........3........3.........1

1........4........6........4.........1

.....1........5.......10......10........5.........1

1........6.......15......20.....15.........6.........1

.....1........7.......21......35......35.......21........7.........1

1........8.......28......56......70......56.......28........8.........1

The central "spine" numbers correspond to the numbers in your question (although you did not include the initial 1). The numbers in this sequence begin:

1, 2, 6, 20, 70, 252, 924, 3432, 12780, 48620, 184756, 705432, ...

They can be calculated directly in various ways. For example, the (n + 1)th number (let's call it Sn+1) is given by the simple formula:

Sn+1 = (2n)! / (n!)2

The problem with this formula is that as n increases, (2n)! grows very quickly, and soon exceeds the limits of even a good pocket calculator. For example, the factorial of 18 is 6402373705728000, which has 16 digits - several more than most calculators can display (although still well within the limits of the Windows calculator).

Luckily, there are other ways of calculating the values. Here, for example, is a solution that involves calculating each new value from the previous one:

S1 = 1;Sn+1 =Sn(4n - 2)/n

Thus, if we begin with S1 = 1, we get the following sequence:

1; x 2/1 = 2; x 6/2 = 6; x 10/3 = 20; x14/4 = 70; etc.

There are many other details I could discuss, such as triangular numbers and Catalan numbers, both of which are related to the series dealt with here. But for now I'll leave things as they stand. If you are interested, you could try dividing successive terms: 2/1, 6/2, 20/6, 70/20, etc. You should soon be able to determine a general rule for how the numbers grow.

* Note that, technically speaking, there are infinitely many possible answers to your question, as there are to any similar question regarding an unfinished sequence. The answer provided here is just the most "obvious" one (i.e., the supposedly "correct" answer expected in an I.Q. test).

Yours,

J. Locke (University of Florence, Italy)

March 2011 - revised and resubmitted after gross vandalism, apparently by a power-mad invigilator, who, perhaps failing to understand the extended answer (if, indeed, he even read it), decided to revert to the first answer given, without even taking the trouble to correct the obvious spelling mistake therein.

John Finch's answer is fine as far as it goes, but it offers very little explanaton. My answer is intended to provide rather more of the mathematical background. If answerman1621 has a better explanation, then he should provide it, rather than just undoing the hard work of someone who has taken the trouble to repond adequately to the question.

Aha! Looking again at John's brief answer, I see that it leaves plenty of room for sponsored links, all visible without scrolling. Suddenly, all becomes clear!

1

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