1, 4, 9, 16, 25, 36, 49, (....)

There are two series (10, 13, 16, 19) and (5, 10, 20, 40), one increasing by 3 and the other multiplied by 2.

Each number is twice the preceding one with 1 added or subtracted alternatively.

So, the next number is (2 x 261 + 1) = 523.

The numbers are squares of odd natural numbers, starting from 5 up to 15.

So, 36 is wrong.

The numbers are 7 x 8, 8 x 9, 9 x 10, 10 x 11, 11 x 12, 12 x 13.

So, 150 is wrong.

The terms are alternatively multiplied by 1.5 and divided by 3. However, 18.5 does not satisfy it.

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

So, the correct next number is 1440.

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

Numbers are all primes. The next prime is 43.

Numbers are alternatively multiplied by 3 and divided by 2.

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

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.