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The image you uploaded appears to be a worksheet titled "ROLL THE CANNONBALLS!" related to counting cannonballs stacked in a pyramid formation. The instructions ask to use a difference table to solve for the number of cannonballs in various pyramid configurations.

From what I can see, the pyramid starts with a base of triangular numbers (3 balls for the first pyramid, 10 balls for the second, etc.), and the questions ask you to analyze the pattern in the sequence of these stacked cannonballs.

Here’s how the questions seem structured:

1. **Use a difference table to find the number of cannonballs in the sixth and seventh pyramids.**  
   - A difference table helps by identifying the change in number between consecutive terms of a sequence. You can use this method to predict future terms, such as for the 6th and 7th pyramids.

2. **What is the number of cannonballs in the 8th pyramid in the sequence?**
   - After constructing the difference table, you'll apply it to find the number of cannonballs for the 8th pyramid.

3. **What is the number of cannonballs in the \( n^{th} \) pyramid in the sequence?**  
   - This involves creating a conjecture based on the pattern identified from previous terms. You will likely need to establish a general formula for the number of cannonballs in the \( n^{th} \) pyramid.

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If you'd like, I can help you set up the difference table or derive the general formula. Let me know how you'd like to proceed!

Would you like me to walk you through constructing the difference table or provide a detailed solution?

### Here are 5 related questions to expand this topic:
1. What is a difference table, and how does it help in identifying patterns in sequences?
2. Can you describe the relationship between triangular numbers and the number of cannonballs in these pyramids?
3. How can you generalize the number of cannonballs in a pyramid with a triangular base of side length \( n \)?
4. What are higher-order difference tables, and when are they useful?
5. How can you verify your results for the number of cannonballs in the 8th pyramid using an alternative method?

**Tip**: The formula for triangular numbers can be very useful in these kinds of problems: \( T_n = \frac{n(n+1)}{2} \). This will help you calculate the number of cannonballs at different pyramid levels.

Use a difference table to solve the number of cannonballs in the sixth pyramid and in the seventh pyramid. What is the number on the eighth pyramid in the sequence? What is the number on the n-th pyramid in the sequence?

Learn how to calculate the number of cannonballs in a stacked pyramid using a difference table. Discover the pattern in triangular numbers and solve for the sixth, seventh, and eighth pyramids. Suitable forstudents in Grades 7-10, this activity focuses on sequences, triangular numbers, and inductive reasoning.

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