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.
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
The series alternates between two patterns: one where numbers are multiplied by 3 and then reduced (18 to 27 to 36) and another where numbers are multiplied by 4 and then reduced (81 to 72 to 63). Following this pattern, the next number after 36 (from the first sequence) would be 45 (36 multiplied by 1.25), and after 63 (from the second sequence), the next number would be 54 (63 reduced by 9). Thus, the next number in the series is 45.
The sequence alternates between multiplying a number and then adding one. Starting with 7, multiplying by 3 gives 21, then adding 1 gives 8. Next, multiplying 8 by 9 gives 72, and adding 1 gives 9. Following this pattern, the next step would be to multiply 9 by 11, resulting in 99. Thus, the next number is 99.
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
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.
The sequence alternates between multiplying a number and then adding one. Starting with 7, multiplying by 3 gives 21, then adding 1 gives 8. Next, multiplying 8 by 9 gives 72, and adding 1 gives 9. Following this pattern, the next step would be to multiply 9 by 11, resulting in 99. Thus, the next number is 99.