Square Pattern Analysis: Locating the 16-Square Structure in a Geometric Sequence

Q: How do I know this follows the n² pattern?

Look at the sequence: Structure 1 has 1 square (1²), Structure 2 has 4 squares (2²), Structure 3 has 9 squares (3²). Each structure forms a perfect square grid!

Q: What if I can't see the pattern clearly in the diagram?

Start with the simplest structures first. Count squares in Structure 1 and 2, then look for the mathematical relationship. The pattern $ n^2 $ will become obvious!

Q: Why is the answer 4 and not 16?

The question asks which structure (position number) has 16 squares, not how many squares total. Structure 4 is the 4th in the sequence and contains 16 squares.

Q: How can I verify this without counting every square?

Use the formula! If Structure 4 has $ 4^2 = 16 $ squares, and you can see it's a 4×4 grid, then 4 × 4 = 16 confirms your answer.

Q: What would Structure 5 look like?

Structure 5 would be a 5×5 grid containing $ 5^2 = 25 $ squares. This pattern continues: each structure n contains n² squares in an n×n arrangement.

Square Numbers with Geometric Pattern Recognition

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 16 squares?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Will there be a term with 16 squares? If so, at which position?

00:14 Let's count the squares in each term, step by step.

00:33 Notice, the number of squares equals the term's position squared.

00:39 So, this is our sequence formula: position, squared.

00:46 Now we need to find if there's a term with 16 squares.

00:51 Let's substitute 16 in the formula, and solve for N.

00:55 We'll take the square root of 16 to find N.

00:59 Remember, N must be positive; sequences don't have negative positions.

01:04 And that's how we solve the question. Well done!

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 16 squares?

Step-by-step solution

To solve this problem, we need to recognize the sequence formed by the given structures of squares.

First, observe the pattern:
- Structure 1: 1 square
- Structure 2: 4 squares (a 2x2 grid)
- Structure 3: 9 squares (a 3x3 grid)
- Structure 4: 16 squares (a 4x4 grid)

The number of squares in each structure corresponds to square numbers: 1, 4, 9, 16, etc. These numbers are significant as they follow the pattern $n^2$ where $n$ represents the position of the structure in the sequence.

Next, let's apply the pattern:

Step 1: Recognize that the number of squares in each structure is given by $n^2$ .
Step 2: We need to find an $n$ such that $n^2 = 16$ .
Step 3: Solving $n^2 = 16$ , we find $n = 4$ .

Thus, the structure with 16 squares is the 4th element in the sequence.

Therefore, the correct answer is $4$ .

Final Answer

$4$

Key Points to Remember

Essential concepts to master this topic

Pattern Rule: Each structure contains n² squares where n is position number
Technique: Find n where n² = 16, so n = 4
Check: Structure 4 has 4² = 16 squares in 4×4 grid ✓

Common Mistakes

Avoid these frequent errors

Counting individual squares instead of recognizing the pattern
Don't try to count every single square in each structure = wastes time and leads to errors! This approach becomes impossible with larger structures. Always look for the underlying pattern: structure n contains n² squares.

Practice Quiz

Test your knowledge with interactive questions

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

FAQ

Everything you need to know about this question

How do I know this follows the n² pattern?

Look at the sequence: Structure 1 has 1 square (1²), Structure 2 has 4 squares (2²), Structure 3 has 9 squares (3²). Each structure forms a perfect square grid!

What if I can't see the pattern clearly in the diagram?

Start with the simplest structures first. Count squares in Structure 1 and 2, then look for the mathematical relationship. The pattern $n^2$ will become obvious!

Why is the answer 4 and not 16?

The question asks which structure (position number) has 16 squares, not how many squares total. Structure 4 is the 4th in the sequence and contains 16 squares.

How can I verify this without counting every square?

Use the formula! If Structure 4 has $4^2 = 16$ squares, and you can see it's a 4×4 grid, then 4 × 4 = 16 confirms your answer.

What would Structure 5 look like?

Structure 5 would be a 5×5 grid containing $5^2 = 25$ squares. This pattern continues: each structure n contains n² squares in an n×n arrangement.

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