Let's go through each sequence to predict the next term using the difference table method. We will calculate the differences between consecutive terms and use this pattern to find the next term.

1) Sequence: 2, 6, 22, 56, 114, 202, _____

• Differences between terms:
  6 - 2 = 4
  22 - 6 = 16
  56 - 22 = 34
  114 - 56 = 58
  202 - 114 = 88

• Second differences:
  16 - 4 = 12
  34 - 16 = 18
  58 - 34 = 24
  88 - 58 = 30

• Third differences (constant):
  18 - 12 = 6
  24 - 18 = 6
  30 - 24 = 6

Since the third difference is constant, we add 36 to 88 to find the next difference:
88 + 36 = 124

Now, add this difference to the last term:
202 + 124 = 326

The next term is 326.

2) Sequence: 3, 9, 15, 21, 27, _____

• Differences between terms:
  9 - 3 = 6
  15 - 9 = 6
  21 - 15 = 6
  27 - 21 = 6

Since the first differences are constant, this is an arithmetic sequence with a common difference of 6.
The next term will be:
27 + 6 = 33

3) Sequence: 2, 6, 12, 20, 30, 42, _____

• Differences between terms:
  6 - 2 = 4
  12 - 6 = 6
  20 - 12 = 8
  30 - 20 = 10
  42 - 30 = 12

The difference increases by 2 each time.
The next difference will be:
12 + 2 = 14

Now, add this to the last term:
42 + 14 = 56

4) Sequence: 6, 11, 18, 27, 38, 51, _____

• Differences between terms:
  11 - 6 = 5
  18 - 11 = 7
  27 - 18 = 9
  38 - 27 = 11
  51 - 38 = 13

The difference increases by 2 each time.
The next difference will be:
13 + 2 = 15

Now, add this to the last term:
51 + 15 = 66

5) Sequence: 1, 8, 21, 40, 65, 96, _____

• Differences between terms:
  8 - 1 = 7
  21 - 8 = 13
  40 - 21 = 19
  65 - 40 = 25
  96 - 65 = 31

The difference increases by 6 each time.
The next difference will be:
31 + 6 = 37

Now, add this to the last term:
96 + 37 = 133

Let me know if you'd like further details on any step! Here are 5 related questions:

1. What is the general formula for an arithmetic sequence?
2. How do we determine if a sequence is quadratic based on differences?
3. What are higher-order differences in sequences and their importance?
4. Can all sequences be predicted using difference tables?
5. How can geometric sequences be handled differently from arithmetic ones?

Tip: Sequences with constant third differences often indicate a cubic polynomial relationship!