15, 31, 63, 127, 255, (....)

Numbers are alternatively multiplied by 2 and increased by 3.

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

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.

Each number is the proceeding number multiplied by -2.

So, the required number is -128.

The pattern is 1 x 2, 2 x 3, 3 x 4, 4 x 5, 5 x 6, 6 x 7, 7 x 8.

So, the next number is 8 x 9 = 72.

Numbers are alternatively multiplied by 3 and divided by 2.

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

Numbers are 1³, 2³, 3³, 4³, 5³, 6³.

So, the missing number is 7³ = 343.

There are two series, beginning respectively with 3 and 7. In one 3 is added and in another 2 is subtracted.

The next number is 1 - 2 = -1.

Let the given numbers be A, B, C, D, E, F, G.

Then, A, A x 1 + 1, B x 2 + 2, C x 3 + 3, D x 4 + 4, E x 5 + 5, F x 6 + 6 are the required numbers.

Clearly, 228 is wrong.

Go on multiplying with 1, 2, 3, 4, 5, 6 to get next number.

So, 96 is wrong.

Numbers must be (14)², (13)², (12)², (11)², (10)², (9)², (8)².

So, 80 is wrong.