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Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 5 such that the first five numbers are as listed in the question followed by the chosen next number. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

In this case the simplest solution, based on a polynomial of order 4 is

t(n) = (-77*n^4 + 874*n^3 - 3400*n^2 + 5321*n - 2676)/6 for n = 1, 2, 3, ... and accordingly, the next number is 693.

Q: What is the next number in the series of 7 21 8 72 9?

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The next number in the series 3, 618, 72, is -1635. The relevant rule is Un = (-1161n2 + 4713n - 3546)/2 for n = 1, 2, 3, ...

Each number in the series is being multiplied by (2 + (n-1)) where n = the term in the series 3 x 2 = 6 6 x 3 = 18 18 * 4 = 72 72 x 5 = 360

360 each number is multipled by (2 + the preceding term number) 3 x 2 = 6 6 x 3 = 18 18 x 4 = 72 72 x 5 = 360

3 can go into both these numbers 72/3= 24 21/3= 7

These are squares. 12 = 1 22 = 4 32 = 9 (it is missing 42 = 16) 52 = 25 62 = 36 The next one in the series is probably 72 = 49.

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If you look at this series, you can see that each number is 2/3 of the one before it. The next number then is 2/3 * 32, which equals 64/3, or 21 1/3.

7, 21, 8, 72, 9, ?

25

32

The next number in the series 3, 618, 72, is -1635. The relevant rule is Un = (-1161n2 + 4713n - 3546)/2 for n = 1, 2, 3, ...

132

243 is one possibility.

45 54

Each number in the series is being multiplied by (2 + (n-1)) where n = the term in the series 3 x 2 = 6 6 x 3 = 18 18 * 4 = 72 72 x 5 = 360

3 * 2 = 6 * 3= 18 *4 =72 *5 = 360

364.5 Each time, you're adding half of the previous number.

could be 72