https://file.aiquickdraw.com/mathbot/file/172770599479868uqs77i.jpg

In the image, problem 5 asks you to write a recursive definition for the sequence \( Q(n) \), given some values of \( Q \) in a table:

\[
\begin{array}{|c|c|}
\hline
n & Q(n) \\
\hline
1 & 3 \\
3 & 15 \\
7 & 39 \\
\hline
\end{array}
\]

### Approach:
1. From the values provided, we need to identify a pattern or rule that generates the next term based on previous ones.
2. Once we find that rule, we can express the sequence recursively.

### First Differences:
Calculate the differences between consecutive terms of \( Q(n) \):
- \( Q(3) - Q(1) = 15 - 3 = 12 \)
- \( Q(7) - Q(3) = 39 - 15 = 24 \)

This suggests that the differences between consecutive terms are increasing by 12, so we have a second-order recursive relation.

### Recursive Formula:
The recursive definition can be built using the fact that the difference between consecutive terms grows by a constant amount:
- \( Q(n+1) = Q(n) + 12 \), with initial conditions based on \( Q(1) = 3 \).

Now that we've identified the relation, let's express it more formally.

Would you like more details or have any questions on this problem? Here are 5 follow-up questions you can explore:

1. How do second differences relate to sequences?
2. How can you extend this sequence beyond \( n = 7 \)?
3. What is the general form of an arithmetic sequence and how does it apply here?
4. Can this sequence be represented in closed form, not just recursively?
5. How does recursion differ from direct formulas in sequences?

**Tip:** When dealing with recursive sequences, always check if you can express the next term as a function of the previous terms for better understanding!

Write a recursive definition for the sequence Q based on the given values.

This problem explores the recursive definition for a sequence Q where the differences between consecutive terms increase by 12. Learn how to calculate the recursive formula for sequences and apply the arithmetic progression concepts. Suitable for Grades 8-10 students.

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