To solve for the 6th term ($p$) and 7th term ($q$) of the sequence $8, 12, 29, 42, \cdots$, using the method of second-order differences, we'll follow these steps:

Step 1: Compute the first differences.

The first difference is the difference between consecutive terms in the sequence.

$\text{First difference} = (12 - 8), (29 - 12), (42 - 29), \cdots$

$\Rightarrow 4, 17, 13, \cdots$

Step 2: Compute the second differences.

The second difference is the difference between consecutive values of the first differences.

$\text{Second difference} = (17 - 4), (13 - 17), \cdots$

$\Rightarrow 13, -4, \cdots$

Step 3: Assume the second differences are constant.

We are told that the second differences remain constant, so we assume the second difference continues to be $-4$.

Thus, the second differences are: $13, -4, -4, \cdots$.

Step 4: Extend the first differences.

To find the next first differences, we add the second differences to the existing first differences:

• After $13$, add $-4$: $13 + (-4) = 9$
• After $9$, add $-4$: $9 + (-4) = 5$

Thus, the first differences become: $4, 17, 13, 9, 5$.

Step 5: Extend the sequence.

To find the 6th and 7th terms, we add the first differences to the current sequence values:

• The 5th term (after 42) is: $42 + 9 = 51$
• The 6th term ($p$) is: $51 + 5 = 56$

Thus, the 6th term is $p = 56$.

Now, for the 7th term ($q$):

• The next first difference is $5 + (-4) = 1$.
• The 7th term ($q$) is: $56 + 1 = 57$

Thus, the 7th term is $q = 57$.

Final Answer:

• $p = 56$ (the 6th term)
• $q = 57$ (the 7th term)

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Would you like further details or have any questions?

Here are 5 related questions:

1. How can second-order differences be applied to quadratic sequences?
2. What if the second-order differences were not constant? How would that affect the solution?
3. Can you explain the relationship between first, second, and third-order differences?
4. How can you use this method for sequences with fractional terms?
5. What are some real-world applications of difference methods?

Tip: Second-order differences suggest that the underlying rule of the sequence may be quadratic in nature.