2, 6, 12, 20, 30, 42, 56, (....)

Numbers are alternatively multiplied by 3 and divided by 2.

So, the next number = 54 ? 2 = 27.

Numbers are all primes. The next prime is 43.

There are two series (8, 11, 14, 17, 20) and (7, 12, 17, 22) increasing by 3 and 5 respectively.

Go on multiplying the given numbers by 2, 3, 4, 5, 6.

So, the correct next number is 1440.

Numbers are 1², 2², 3², 4², 5², 6², 7².

So, the next number is 8² = 64.

Each number is the proceeding number multiplied by -2.

So, the required number is -128.

Numbers are (2³ - 1), (3³ - 1), (4³ - 1), (5³ - 1), (6³ - 1), (7³ - 1) etc.

So, the next number is (8³ - 1) = (512 - 1) = 511.

Numbers are alternatively multiplied by 2 and increased by 3.

So, the missing number = 61 x 2 = 122.

Each number is double the preceding one plus 1.

So, the next number is (255 x 2) + 1 = 511.

Numbers are 1³, 2³, 3³, 4³, 5³, 6³.

So, the missing number is 7³ = 343.