The pattern is to subtract 8, then divide by 2. 

The last two given numbers in the series are 66 and 33. 

Since \(\displaystyle \frac{66}{2}=33\),

the next operation is to subtract 8.

\(\displaystyle 33-8=25\)

The pattern is multiply by 10, then divide by 2. 

\(\displaystyle 7\cdot 10 = 70\)

\(\displaystyle \frac{70}{2} = 35\)

\(\displaystyle 35 \cdot 10 = 350\)

\(\displaystyle \frac{350}{2} = 175\)

\(\displaystyle 175 \cdot 10 = 1,750\)

The pattern has three parts:  add 4, subtract 7, add 13. 

The sequence has gone through one whole cycle and two-thirds of the second. 

Therefore, the next part of the pattern is to add 13 to the last number.  

\(\displaystyle 61 + 13 = 74\)

There are two different ways to solve this problem.  The first pattern is to divide by 3, then multiply by 6. 

This would mean we should divide 2448 by 3.

\(\displaystyle \frac{2448}{3}=816\)

The second way to view this pattern is that there are two groups of numbers which double themselves.  If we think of 612, 1224 and 2448 as the A sequence, each number in this sequence is multipled by 2.  Think of 204, 408 and the unknown number we're looking for as the B sequence.  408 is double 204.  It is safe to assume that the next number in the B sequence would be double 408. 

\(\displaystyle 408 \cdot 2 = 816\)

This sequence has groupings of three numbers.  The three numbers in the first group are 3, 4 and 5.  The three numbers in the second group are 9, 12 and 15.  The first number in the third group is 27.  How then do we determine the second number in the third group?  By placing the first group in a line above the second group, and the second group above the third group, we can determine their relationships.

Group 1:     3      4      5

Group 2:     9     12     15

Group 3:     27    ?       ?

Based on the first numbers in each group (3, 9 and 27), we can determine that each group of numbers have been multiplied by 3.  Therefore if we multiply 12 by 3, we get 36. 

The pattern is to subtract .6 and then invert the next number.  (To invert a number, you merely reverse the digits.)

\(\displaystyle 9.1 - 0.6 = 8.5\)

5.8 is the inverse of 8.5

\(\displaystyle 5.8 - 0.6 = 5.2\)

2.5 is the inverse of 5.2

\(\displaystyle 2.5 - 0.6 = 1.9\)

The pattern is to subtract 4. 

The number 100 appears every 4th number in the regular pattern. 

If we ignore the number 100, the last operation should be:

\(\displaystyle 87 - 4 = 83\)

The pattern is to subtract 13 from each number.

\(\displaystyle 739 - 13 = 726\)

The pattern is add 57, then subtract 32. 

\(\displaystyle 47 + 57 = 104\)

\(\displaystyle 104 - 32 = 72\)

\(\displaystyle 72 + 57 = 129\)

\(\displaystyle 129 - 32 = 97\)

\(\displaystyle 97 + 57 = 154\)

This sequence is simply a list, in order, of prime numbers beginning with 109. 

A prime number is a number which can only be divided evenly by itself and 1.