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How would you find the number of squares at stage n

To find the number of squares at stage \( n \) in the sequence of images, we can observe the pattern of growth across the stages:

1. **Stage 1**: There is 1 square.
2. **Stage 2**: There are 3 squares.
3. **Stage 3**: There are 6 squares.
4. **Stage 4**: There are 10 squares.

The number of squares at each stage corresponds to the sum of the first \( n \) natural numbers. This sum can be calculated using the formula:

\[
S_n = \frac{n(n+1)}{2}
\]

where \( S_n \) is the number of squares at stage \( n \).

- For **Stage 1**: \( S_1 = \frac{1(1+1)}{2} = \frac{2}{2} = 1 \)
- For **Stage 2**: \( S_2 = \frac{2(2+1)}{2} = \frac{6}{2} = 3 \)
- For **Stage 3**: \( S_3 = \frac{3(3+1)}{2} = \frac{12}{2} = 6 \)
- For **Stage 4**: \( S_4 = \frac{4(4+1)}{2} = \frac{20}{2} = 10 \)

So, the number of squares at stage \( n \) is given by:

\[
S_n = \frac{n(n+1)}{2}
\]

Would you like further details or explanations?

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Here are some related questions you might find interesting:

1. How does the sum of the first \( n \) natural numbers relate to the number of squares?
2. Can you derive a recursive formula for the number of squares at each stage?
3. What would be the number of squares at stage 10?
4. How can you represent the relationship between \( n \) and the number of squares graphically?
5. How does this pattern relate to triangular numbers?
6. Is there a way to generalize this pattern for other types of shapes?
7. What is the difference between this pattern and an arithmetic progression?
8. How could this pattern be applied in real-world scenarios?

**Tip:** Understanding patterns like these is fundamental to solving problems in both arithmetic and geometric progressions, which are key concepts in algebra.

Learn how to calculate the number of squares at stage n in a sequence of images. This problem involves understanding arithmetic sequences and the summation of series formula. Suitable for students in Grades 6-8.

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