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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**

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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!

Use the difference table to predict the next term in the sequence.

Learn how to use difference tables to predict the next term in a sequence. We explore arithmetic and polynomial sequences, examining first and second differences to find patterns. Suitable for students in Grades 8-10.

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