Finding the 81-Square Element in a Geometric Square Sequence

Q: How do I know this is about square numbers?

The key clue is that we're looking for 81 squares in a geometric pattern. Since 81 is a perfect square (9²), we're dealing with square number sequences where each structure contains n² unit squares.

Q: What if I don't remember that 81 is a perfect square?

No problem! Just set up the equation $ n^2 = 81 $ and solve by taking the square root of both sides. You can also try small values: 8² = 64, 9² = 81, 10² = 100.

Q: Why is the answer 9 and not 81?

The question asks for which structure number contains 81 squares, not how many squares there are. Structure 9 is a 9×9 grid containing 81 unit squares.

Square Sequences with Perfect Square Patterns

The following is a series of structures formed by squares with side lengths of 1 cm.

In which structure (element) of the series are there 81 squares?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Will there be a term with 81 squares? If so, at what position?

00:03 Let's count the squares in each term

00:24 We can see that the number of squares equals the term's position squared

00:31 Therefore we can conclude this is the sequence formula

00:39 We want to find if there's a term with 81 squares

00:44 Let's substitute in the formula and solve for N

00:48 Let's take the square root to isolate N

00:53 N must be positive, there's no negative position in the sequence

00:58 And that's the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The following is a series of structures formed by squares with side lengths of 1 cm.

In which structure (element) of the series are there 81 squares?

Step-by-step solution

To solve this problem, let's consider the sequence structure for square numbers. We are tasked with finding the structure that contains 81 squares, implying a perfect square sequence. Therefore, we need to identify the correct term that expresses this number of squares directly.

Step 1: Recognize that each structure corresponds to an $n \times n$ arrangement.
Step 2: Use the formula for square numbers: $n^2$ .
Step 3: Set up the equation $n^2 = 81$ .

Solving for $n$ :

n^2 = 81

Taking the square root of both sides gives:

n = \sqrt{81} = 9

Thus, the structure in which there are 81 squares is the 9th structure in the sequence.

Therefore, the solution to the problem is $n = 9$ .

Final Answer

$9$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Each structure forms an n×n grid of unit squares
Formula Application: Use n² = 81, so n = √81 = 9
Verification: Check that 9² = 9 × 9 = 81 squares total ✓

Common Mistakes

Avoid these frequent errors

Counting individual squares instead of recognizing the pattern
Don't try to count each small square in the diagram = waste time and make errors! This leads to miscounting and confusion. Always recognize that structure n contains n² unit squares in an n×n arrangement.

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

How do I know this is about square numbers?

The key clue is that we're looking for 81 squares in a geometric pattern. Since 81 is a perfect square (9²), we're dealing with square number sequences where each structure contains n² unit squares.

What if I don't remember that 81 is a perfect square?

No problem! Just set up the equation $n^2 = 81$ and solve by taking the square root of both sides. You can also try small values: 8² = 64, 9² = 81, 10² = 100.

Why is the answer 9 and not 81?

The question asks for which structure number contains 81 squares, not how many squares there are. Structure 9 is a 9×9 grid containing 81 unit squares.

How can I visualize this pattern?

Think of it like this: Structure 1 has 1² = 1 square, Structure 2 has 2² = 4 squares in a 2×2 grid, Structure 3 has 3² = 9 squares in a 3×3 grid, and so on.

What if the number wasn't a perfect square?

If we needed a non-perfect square like 80, there would be no exact structure in this sequence. The sequence only contains perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

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