Look at the pattern below. step 1 with 1 square, step 2 with 3 squares, step 3 with 6 squares, step 4 with 10 squares. How does the pattern grow at each step?
Understand the Problem
The question is asking to identify the growth pattern of the number of squares in each step, based on the provided steps and their respective square counts. We need to analyze the differences between successive steps to determine how the amount of squares increases.
Answer
The number of squares follows the pattern $n^2$.
Answer for screen readers
The number of squares at each step follows the pattern $n^2$, where $n$ is the step number.
Steps to Solve
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Identify the sequence of squares Start with listing the number of squares from each step, for example: Step 1: 1 square, Step 2: 4 squares, Step 3: 9 squares, Step 4: 16 squares, … and so on.
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Look for a pattern in the squares Recognize that the sequence of squares resembles perfect squares, which are defined as the square of whole numbers. Thus, we can represent the number of squares at step $n$ as $n^2$.
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Express the growth mathematically The growth pattern can be shown with the formula for the number of squares: $$ \text{Number of squares} = n^2 $$ where $n$ represents the step number.
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Find the differences between successive steps Calculate the difference in the number of squares between each successive step:
- From Step 1 to Step 2: $4 - 1 = 3$
- From Step 2 to Step 3: $9 - 4 = 5$
- From Step 3 to Step 4: $16 - 9 = 7$
- Identify the pattern in the differences Notice that the differences form an arithmetic pattern where each subsequent difference increases by 2:
- 3, 5, 7...
The number of squares at each step follows the pattern $n^2$, where $n$ is the step number.
More Information
The sequence of square numbers (1, 4, 9, 16, ...) corresponds to the squares of integers (1, 2, 3, 4, ...). This pattern continues indefinitely, representing a quadratic growth.
Tips
A common mistake might be to overlook the square pattern and instead look for a linear relationship where none exists. Always check for known number sequences.
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